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Introduction Proposition 1.1 Suppose G is chordal. Then an edge-coloring of G forbids rainbow cycles iff it forbids rainbow K3 s.xx / , 4R<PF> () Clear. () Claim  If an edge-coloring of G has a rainbow cycle, then It admits a rainbow K3. Now let C be a shortest rainbow cycle with |C|4. )0 $yTheorem JL If the edges of Kn are colored with n or more colors actually appearing, then there is a rainbow K3 somewhere.z  <  "] 8*JL(n) coloring A JL(n) coloring is a coloring of the edges of Kn with exactly n-1 colors, forbidding rainbow K3 s.t*  6 *3 ' Characterization of JL(n) Colorings D(" x "  jLemma 2.1 Suppose n 2 and Kn is colored with a JL(n) coloring. Then "vV(Kn), there is exactly one color appears on edges incident to v and that color doesn t appear in Kn -v.zx     $" P ^      <PF of Lemma 2.1> By Theorem JL, at least one color appears on the edges incident to v which is not in Kn-v. (o.w. $ a rainbow K3 in Kn-v.) If more than two colors appears on the edges incident to v which is not in Kn-v, then $ a rainbow K3.   7"PhR "Corollary 2.2 Suppose S V(Kn), 1 |S| n, and Kn is colored with a JL(n) coloring. Then the coloring restricted to <S> is a JL(|S|) coloring.vn- >G<PF> If |S|=n, then it is clear. If |S|<n, let v1, v2, ..., vk be some ordering of V(Kn)-S. Apply Lemma 2.1 k times.lvn     ,< 9Corollary 2.3 In a JL(n) coloring no K3 is monochromatic.:x #V By corollary 2.2, let |S|=3. JL(3) coloring is a coloring with exactly 2 colors.#4    9Theorem 2.4 Every JL(n) coloring is obtained as follows: ^:n $$   r First, if a coloring is obtained by above directions, then the coloring is a JL(n) coloring.ns Q c   Now suppose we have a JL(n) coloring. We want to show that it must be achievable as described above. By induction on n. When n=2, clear. Suppose that n>2. Let vV(Kn). By Lemma 2.1, the coloring restricted to Kn-v is a JL(n-1) coloring. f((f]   >,  By induction hypothesis, $ r0, s0N s.t. r0+s0=n-1 and a partition of V(Kn-v) into two sets R0, S0 with |R0|=r0, |S0|=s0.{ ,$/  By Lemma 2.1, let red be the color on an edge incident to v which does NOT appear in Kn-v. WLOG, red appears on a v-to-R0 edge. &V' 3Then all v-to-S0 edges must be either red or green.~4     VCase 1 All v-to-S0 edges are green. Notice that no v-to-R0 edge is green. Also, no v-to-R0 edge has a color appearing in <S0>. Now let R=R0*"{v}, S=S0. Thus we have done.            O A JL(9) coloring obtained by Thm 2.4 corresponds a full binary tree.BP "'&  ,Essentially the same, essentially different "-+0Two edge-colorings of Kn are essentially the same if there is a permutation of the vertices of Kn and a renaming of the colors that takes one coloring into the other. Otherwise, the two colorings are essentially different .  / g,ICorollary 2.5 Suppose n>1. For any color appearing in a JL(n) coloring, the subgraph of Kn induced by the edges of that color is a complete bipartite graph. The subgraph is spanning in Kn for exactly one color.n!_b8H By Thm 2.4.B Corollary 2.6 For n>2, the essentially different JL(n) colorings are in natural one-to-one correspondence with different full binary trees with n leafs.  [1e ! ACorollary 2.7 For n>1, if n is odd then and if n is even then 4  " <Example> n=9 t9 = t1t8 + t2t7 + t3t6 + t4t5 n=8 t8 = t1t7 + t2t6 + t3t5 + + t4 \ & Frequency"  bIn any coloring of edges, the frequency of a color is the number of edges the color appears on. 0c <' =3. Equalized Edge-colorings of Kn Forbidding Rainbow Cycles>> (( ((  (!  )" <PF> By induction on n. When n=2, it is clear. Suppose that n>2 and Kn is edge-colored with no rainbow K3 s.Pi " D(*# Let vV(Kn). In Kn-v, $ a connected spanning subgraph induced by green.(by induction hypothesis)c-P +$ Let uV(Kn)-{v}, and suppose the edge uv is red. Claim the edges incident to v are all red. 1n,, 5,% :<PF> When n=2, 3, it is clear. For n>4, since It means no color can appear more than n-2 times. By Thm 3.1, there must be one color appearing n-1 times. P2=Z2 % d7.& <PF> () By Cor 3.2. () For t =1, 2, it is true. So assume 3 t There is a JL(n) coloring with frequency vector (n-1, n-2, & , 1). n "# , @60' ^Partition {1, 2, & , n-1} into t sets C1, C2, & , Ct, s.t. . Notice that in an equalized edge-coloring, every color must appears either or times. Now, merging the colors in the above JL(n) coloring whose frequency is in Cj into a single color. 522*,2Zn4c%">5"3( Greedy method Assume m1 m2 & mt. We will pick the elements of C1, then C2,etc.. Start the picking for Cj with the largest unpicked integer which does not cause the sum over Cj to exceed mj.v F P#GF 6) = For n =15, t =6. Then m1=m2=m3=17 and m4=m5=m6=18.$>       ;+ Partition {1, 2, 3,& , 8} into five sets: C1={7}, C2={6,1}, C3={5,2}, C4={4,3}, C5={8}Xn*=-84. Edge-colorings of Kn() \x(0 <,Lemma 4.1 In any rainbow-K3-forbidding edge coloring of Kn, any two edges bearing colors of frequency 1 in the coloring must be non-incident. zx   S8U ">.xCorollary 4.2 In any rainbow-K3-forbidding edge coloring of Kn, no more than of the colors have frequency 1. zy  9<; A/VLemma 4.3 There is a JL(n) coloring in which of the colors have frequency 1.TWn  >>D<PF> If n=1, 2, then it is clear. For n3, partition n into 2, n-2, then partition n-2 into 2, n-4, and so on. Thus there will be nodes with weight 2."< AC0  , K4uv edge bundle , If u and v are distinct vertices in Kn, we refer to the  edges in Kn() incident to u and v as the uv edge bundle. "y " "  *  *  "  "  "  "  "  &   &  "  " >$D1JL(n, ) coloring A rainbow-K3-forbidding edge coloring of Kn() can be obtained from the followings:f&&&&& & ""*""**%"&:4' L3(3) If edge uv bears a color with frequency 1, then color all uv edge bundle in Kn() with  different color.nn ""0""""**"&> 0 H2  A JL(5, 3) coloringXP5Theorem 4.4 The greatest number of colors appearing in an rainbow-K3-forbidding edge coloring of Kn() is n-1+(-1) , and if n, >1, the only such colorings with that number of colors are JL(n, ) colorings.@ .""*""**"""<""""",a` R6 N<PF> Consider  fixed. By induction on n. When n=2, it is clear. Suppose that n>2 and Kn() is edge-colored with no rainbow K3 s using a maximum number of colors. L22|  " "  *  * "   ""XKS7 8Let vV(Kn()) and K = Kn()-v By induction hypothesis, there are no more than n-2+(-1) colors in K. Note that Kn() can be colored with n-1+(-1) colors. So there must be at least one edge incident to v bearing a color NOT in K. Call such colors  new colors . (2M2"** "  *  "  "  *  * """ ""**""="" &"> dV8 ]There cannot be two different new colors appearing in different edge bundles incident to v.&^[W9 Case (a) New colors appear in at least two different edge bundles. (In fact, there is only one new color in this case.) Case (b) New colors appear in only one edge bundle.b: )+X: Case (a) Notice that [n-1+(-1) ] - [n-2+(-1) ] = So in case (a), n must be odd and K must be colored with a JL(n-1, ) coloring. That means every vertex in K is incident to an unique edge bundle which has  different colors. H2Vb"""""""""")""D"[;  If the only one new color (say, red) appears in a uv edge bundle, then $ wV(K)-{u} s.t. every edge in the vw edge bundle is red.  """""P3 \< ?Furthermore, every edge bundle incident to v is monochromatic.,@+]= Kn() Kn monochromatic edge bundle that color -polychromatic edge bundle a new color\"**  "   $R_> Then Kn is edge colored with (n-2)+1=n-1 colors, and = edges are of frequency 1. m *fb? Case (b) New colors appear in only one edge bundle. Let wV(K) and vw be that edge bundle. Then " uV(K)-w, 5 <- F:c@ Note that it must be  different new colors on vw edge bundle. `$ If n is even. Let m be # of colors appearing in Kn() m (# of colors appearing in K) +  (n-2)+(-1) +  = (n-1)+(-1) m = (n-1)+(-1)@ yn""""** """" """ """,/BjC  gB "a$ If n=3, clear. ` fA b$ If n is odd and n5. Consider G= Kn()-v-w Claim there is at most one color on {v, w}-to-G and not in G. So that m (# of colors appearing in G) +1 +  (n-3)+(-1) + (-1)+2 = (n-1)+(-1) m = (n-1)+(-1) ->(0Wn  "**$ &&&&&&&&&&&"! tG=Claim there is at most one color on {v, w}-to-G and not in G.>$ Recall " uV(G), uv and uw are the same color. If green is the color appears on uv and uw for some uV(G) and not in G, then for any xV(G),0(_      8 sF Kn() Kn monochromatic edge bundle that color -polychromatic edge bundle a new color\"**  "    $RuH  vI#Cyclic triple and Transitive triple$$ RUHow many colors can be used to color the arcs of Dn with no rainbow directed triples?<V1 #1#.Consider the rainbow-K3-forbidding edge coloring of Kn() with =2. So the answer is n-1+ , and the only such coloring is a JL(n, 2) coloring. ""*""** "".",4MxJTheorem 5.1 The greatest # of colors in a coloring of the arcs of Dn s.t. there are no rainbow transitive triples is n-1 + .px 6 08B<  t<PF> (Similar to Theorem 4.4) By induction on n. When n=1, 2, it is clear. Suppose that n3 and Dn is arc-colored with no rainbow transitive triples using a maximum number m of colors.,1np  !"*< `Y{K dNotice that m n-1 + , we want to show that m n-1 + . Let vV(Dn) and D=Dn-v. A color on an arc with either end at v but not appear in D will be called a  new color . If a new color appears on (u, v) or (v, u), then no other new color appears on (w, v) or (v, w) for any wV(D). KZ2s """%"              (    ;        (             >PQ `$ There is no new color. By induction hypothesis, m n-2+ +0 n-1 + . a$ There is only one new color. By induction hypothesis, m n-2+ +1 n-1 + . X&"""&"""    "S Vb$ There are two new colors on (u, v) and (v, u). By induction hypothesis, m n-2+ +2 = n-1 + for n is even. So suppose n is odd. Let D=D-u. Claim there is at most one color appear in {u, v}-to-D but not in D. So that m n-3+ +2+1= n-1 + .(!(I5x&&&&&&&&&"""                .           """U Claim there is at most one color appear in {u, v}-to-D but not in D. |J- |LTTheorem 5.2 D is a digraph obtained by orienting the edges of Kn() so that for any pair of distinct vertices there is at least one arc between them in each direction. Then the greatest # of colors in a coloring of the arcs of D s.t. there are no rainbow transitive triples is n-1+(-1) .+x 1"**e<0 "   "   8>A   ~NfTheorem 5.3 Suppose nN and n=4q+r where q, rZ and 0r<4. Let g(n)= Then the arcs of Dn can be colored with g(n) colors appearing so that there are no rainbow cyclic triples.n  @n?   O <PF> `$ For n=3, a$ For n=4, g(3)=4 g(4)=6 Tc""""/"P Db$ For n5. Partition V(Dn) into q 4-sets F1, F2, & , Fq and one r-set R. (R @ D1, D2, D3) Use 6 different colors on each Fi, and 0, 2, or 4 different colors on R @ D1, D2, or D3. If r=0, then use a JL(q) coloring on Fi-to-Fj. If r>0, then use a JL(q+1) coloring on Fi-to-Fj, and Fi-to-R. #"" "&     DM)The End& & .  /   0` zw33fYyy` _,XH` ]]Ri33f` ff3fffT` 33f` 3fY3f` 3VVf{{` }zn3>?" dd@,?vKd@ U v@Z`vU n?" dd@   @@``PR    @ ` ` p>> jb0(    6,Ř "P  L cN NN}/kGrjL#j_    0ǘ "  |0 cN NN}/kGr ,{Nd\ ,{ Nd\ ,{Vd\ ,{Nd\    00Θ "^`  Z*  0 Ԙ "^   \*  0ژ "^   \*l  FA޽h ?1"` zw33fYyy80___PPT10.'ӌ tdesignc  0 >6@(    6qe "p  L cN NN}/kGrjL#j_    0 " `   f N cN NN}/kGroRjL#j_   0 "^`  Z*  0 "^   \*  0 "^   \*n  HA޽h ?1+"` zw33fYyy80___PPT10.'ӌ 0 x`(    0-     2  Z*   0-   H 2  \* d  c $ ?>Mv -   0<< mU -  |0 cN NN}/kGr ,{Nd\ ,{ Nd\ ,{Vd\ ,{Nd\    642     -  Z*   62   H 2  \* H  0~af ? 3380___PPT10.!J `X(  X X 0.2     2  X*  X 002   H 2  Z*  X 672     2  X*  X 6=2   H 2  Z* H X 0~af ? 3380___PPT10.{'K&s 0  d(  x  c $|z )  x  c $t]   ,  <o?"0@NNN?NN  Speaker: 傇[ 8(2  H  0޽h ? 33___PPT10i.!0!`+D=' = @B +3  0 JBp(    N o?"0@NNN?N&j 8___PPT10N___PPT90( F1. Introduction 2. Characterization of JL(n) Colorings 3. Equalized Edge-colorings of Kn Forbidding Rainbow Cycles 4. Edge-colorings of Kn() 5. Forbidding Rainbow Directed Triples in Dn0n2..=n-f..=/H  0޽h ? zw33fYyy___PPT10i.H" M+D='  = @B +  0 (  r  S , }M    H  0޽h ? zw33fYyy___PPT10i.J"(t+D='  = @B +=  0 `X0 :(  Xx X c $    X c $i<$ 0  /l   X&W ,$D  0B X NDԔ?"0@NNN?N + X HJdo?"0@NNN?Nwt   =chord(2* X No?"0@NNN?N &,$  0 @ (2#m l P  XP ,$D  0`B  X 0DoOB X NDo?"0@NNN?N\dB X NDo?"0@NNN?Nzed B X ND33o?"0@NNN?N y B XB NDo?"0@NNN?N. B X NDo?"0@NNN?Nn  B X NDo?"0@NNN?N> en  X T o?"0@NNN?NP ;C(22  X <$   > 2  X <( \U > 2  X <, D d  > 2  X <0 I   > 2 X <43 Y   > 2 X <6    > 2 X <d;  O5 > H X 0޽h ? 9ff3,,___PPT10,.!ā+D+' = = @B D+' = @BA?%,( < +O%,( < +DN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XD! ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X/%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*X/D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*X/D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X/O%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*X/OD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*X/OD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*XOh%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XOhD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XOhDN' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*Xh%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XhD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XhD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*XD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*XD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D ' =%(D9 ' =%(D' =A@BB&BB0B@HB%()?D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =1:B -45.0*k3>'Bstyle.rotation<*X%(DS' =+4 8?lCB4BCB4BBCBB*k3>'Bstyle.rotation<*X%(Dm' =+4 8?CB#ppt_y-1BCBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)B*Y3>B ppt_y<*X%(D' =+[4 8?CBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)BCBY#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)-#ppt_h/2B*Y3>B ppt_y<*X%())?)?Di' =+4 8?CBG#ppt_y-(0.354*#ppt_w-0.172*#ppt_h)BCB#ppt_yB*Y3>B ppt_y<*X%(`+p+0+X0 ++0+X0 +  0 `P(  `x ` c $X C    ` NpZ o?"0@NNN?Ny  ,$ 0 On the other hand, Kn can be edge-colored with n-1 colors appearing and with rainbow K3 s forbidden.e0x2$PH ` 0޽h ? 9ff3___PPT10z.!+vvD' I = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =-6B+checkerboard(across)*<3<*`+8+0+`0 +  0 p`(  `r ` S @_z    H ` 0޽h ? zw33fYyy___PPT10i.)Pi+D=' I = @B +  0 ( (    S Dw 2    H  0޽h ? zw33fYyy___PPT10i.J"`5b+D=' I = @B +  0 d8(  dx d c $P25t<     d N o?"0@NNN?N. Tv,$  0 It follows that the coloring restricted to Kn -v is a JL(n-1) coloring. I0x2++H d 0޽h ? 9ff3___PPT10`.!j)+5 D' I = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-m6Bbox(in)*<3<*d+8+0+d0 +(  0 ('p050 '(  B  NDo?"0@NNN?NmmB  NDo?"0@NNN?NB  NDo?"0@NNN?NmB  NDo?"0@NNN?NB  NDo?"0@NNN?NmB  NDo?"0@NNN?Nm  H8 o?"0@NNN?Ndg JV1,(2   H\ o?"0@NNN?Nt JV2,(2   H o?"0@NNN?Nt JV3,(2   H( o?"0@NNN?Neh  JV4,(2 2  < " 0 > B  TDo?"0@NNN?NB  NDo?"0@NNN?N}}B  TDo?"0@NNN?NPB  NDo?"0@NNN?N/GG}B  TDo?"0@NNN?NG}B  NDo?"0@NNN?NG}  Hȫ o?"0@NNN?Nz a JV1,(2   Hİ o?"0@NNN?N"   JV2,(2   H o?"0@NNN?NP7 JV3,(2   H o?"0@NNN?Nza JV4,(2  " B o?"0@NNN?Nm i >(2 0 # B o?"0@NNN?NS ']  $K4 is colored with a JL(4) coloring.F%(2 F $ B o?"0@NNN?N3 2 K4-v1 (green)d(2  2 ( <    > B * NDo?"0@NNN?Nc 4c B + TDo?"0@NNN?N 4 B , TDo?"0@NNN?Nc  B - NDo?"0@NNN?N bb B . NDo?"0@NNN?Nc b B / TDo?"0@NNN?Nc b  0 H o?"0@NNN?N   JV1,(2  1 H o?"0@NNN?N j  JV2,(2  2 HL o?"0@NNN?N   JV3,(2  3 H o?"0@NNN?N  JV4,(2 R 4 B o?"0@NNN?N9 ` v  K4-v2 (blue) p(2  5 B o?"0@NNN?N S . (22  <, @r > 2  < @I > 2  < r > 2  <l I > 2  <x u0 > 2  <| P > 2  < Pu > 2 & < " 6   > 2 ' < 6 5  > 2 ) <  5  > H  0޽h ? zw33fYyy___PPT10i.!+D=' I = @B +B  0 A9(    S _,<$@ 0 _ B  NDo?"0@NNN?N% C ,$@ 0B  HDo?"0@NNN?Nc C ,$@ 0B  NDo?"0@NNN?N C  ,$@  0B  HDo?"0@NNN?N C ,$@  0   C 0e0e    BCDEFA Ao 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||@8q<r-@  "0e@`     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab B ,$D 0-  H o?"0@NNN?N_ F ,$ 0 IV,(22  <$ 6   ,$@  0 > 2  <(    ,$@  0 > 2  N1?"0@NNN?NtJ ,$@  0  H\- o?"0@NNN?NIFC,$ 0  Kn -vb(22  <4 s   ,$@ 0 > 2  <7  S ,$@ 0 > 2  <;  q ,$@ 0 > H  0޽h ? zw33fYyy0-(-___PPT10-.!p\+9DT+' I = @B D+' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D_' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(Dk' =%(D' =%(D9' =4@BBBB%(D' =/u8CC*Q3> Br<*D9' =4@BBBB%(D' =/u8CC*Q3> Br<* D9' =4@BBBB%(D' =/u8CC*Q3> Br<*++0+ 0 ++0+ 0 ++0+ 0 ++0+0 ++0+0 ++0+0 ++0+0 +  0 P(  r  S ( w      S (  <$ 0   H  0޽h ? zw33fYyy  ___PPT10 .!"Y++ٺ3DF ' I = @B D ' = @BA?%,( < +O%,( < +D8 ' =%(D ' =%(DH' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-w6Bdiamond(out)*<3<*DH' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-w6Bdiamond(out)*<3<*!DH' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*!T%(D' =-w6Bdiamond(out)*<3<*!TDH' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*T]%(D' =-w6Bdiamond(out)*<3<*T]DH' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*]v%(D' =-w6Bdiamond(out)*<3<*]v+8+0+0 +$  0 P(  r  S       S  ) <$@  0   H  0޽h ? zw33fYyy ___PPT10 .!PXL+1;[D ' = @B Ds ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*"D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*"Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#W%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*#WD' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*#W+   0 c[ (  r  S !    2  NX# o?"0@NNN?N: ,$ 0 HbFurther, any edge-coloring of Kn arrived at by following the directions above is a JL(n) coloring.c 23 ,3 F  N. o?"0@NNN?N,@,$  0 \F(1) Choose r, sN satisfying r+s=n.$ 2 ,   N? o?"0@NNN?Nm,$  0 E(2) Partition V(Kn) into two sets R, S satisfying |R|=r, |S|=s.(F 24D  NDM o?"0@NNN?N% ,$  0 Z\(3) Color all R-to-S edges in Kn with  green ./ 2   N[ o?"0@NNN?NS % ,$  0 @(4) Color <R> with a JL(r) coloring and <S> with a JL(s) coloring with disjoint sets of colors, and with  green not appearing in <R>, nor in <S>.V 2  : ,!cH  0޽h ? zw33fYyyZR___PPT102.!p.+pyD' I = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*++0+0 ++0+0 ++0+0 ++0+0 ++0+0 +Ca  0 VN;(  2  NCo?"0@NNN?NQ 9 ,$@ 02  No?"0@NNN?N9 ,$@ 02  No?"0@NNN?N9 ,$@  02  No?"0@NNN?NP 9 ,$D  0B  HDo?"0@NNN?NJ ,$@ 0B  HDo?"0@NNN?NJ ,$@  0B  HDo?"0@NNN?N x ,$D  0B  HDo?"0@NNN?Nx ,$@  0B  HDo?"0@NNN?Nx V ,$@  0B  HDo?"0@NNN?NJ V ,$@  05  B o?"0@NNN?N}:,$  0 W8 25  Bx o?"0@NNN?N ,$  0 W8 2B @ <Do?"0@NNN?N  ,$D 0l  60  o?"6@ NNN?N  ,$@ 0 JL(r) coloringP 2  l  6  o?"6@ NNN?N  ,$ 0 JL(s) coloringP 2  B  <Do?"0@NNN?NV   ,$@ 0 : <\  #q   < ; H  o?"6@ NNN?NIDe,$ 0 RTNo rainbow K3, and 1+(r-1)+(s-1)=r+s-1=n-1+ vK   \z -   x ,$@  02   68 -` > 2   6l -p > 2   64 c -  > B   ND>?"0@NNN?N%  \z -      ,$@  02  6л -` > 2  6 -p > 2  6 c -  > B  ND>?"0@NNN?N%  H  0޽h ? zw33fYyyFF___PPT10mF.! Y\+@ VDaE'  = @B DE' = @BA?%,( < +O%,( < +Df' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D6' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D=' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DJ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =-6B+checkerboard(across)*<3<*;++0+0 ++0+0 ++0+0 ++0+;0 +  0 F>(    S  <$D 0   H  0޽h ? zw33fYyy___PPT10.!l +YD' I = @B D^' = @BA?%,( < +O%,( < +D' =%(D' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*gz%(D' =-u6Bdiamond(in)*<3<*gzD' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*{%(D' =-6B+checkerboard(across)*<3<*{D' =%(DI' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*+6j  0 &((  r  S     2  NCo?"0@NNN?NTQ w ,$@ 02  No?"0@NNN?NTw ,$@ 02  No?"0@NNN?NTw ,$D   02  No?"0@NNN?NTP w ,$@  0B  HDo?"0@NNN?N6 J 6 ,$@  0B  HDo?"0@NNN?N6 J G ,$@  0B  HDo?"0@NNN?NG x G ,$D 0B  HDo?"0@NNN?N6 x G ,$@  0B  HDo?"0@NNN?N6 x ,$@  0B  HDo?"0@NNN?N6 J ,$@ 0F  B o?"0@NNN?N Tb ,$  0 hH 2F  B o?"0@NNN?N6  ,$  0 hH 2B @ <Do?"0@NNN?N:  ,$D 0e  6L  o?"6@ NNN?N  ,,$@ 0 JL(r0) coloringb 2 e  6  o?"6@ NNN?N ,,$ 0 JL(s0) coloringb 2 B  <Do?"0@NNN?N   ,$@ 0l J  (J ,$@  0B !B HDo?"0@NNN?NJ0 % B "B HDo?"0@NNN?NJ0 6 B $ HDo?"0@NNN?N0  % B % HDo?"0@NNN?N0  6 B & HDo?"0@NNN?N0  B 'B HDo?"0@NNN?NJ0 \z -    x ,$@  02   6X -` > 2   6$ -p > 2   6' c -  > B   ND>?"0@NNN?N%  \z -      ,$@  02  6+ -` > 2  6l/ -p > 2  63 c -  > B  ND>?"0@NNN?N%  )l   "   ",$@  02  68  ]  >   B; o?"0@NNN?N\  " Gv* 2H  0޽h ? zw33fYyyJJ___PPT10J."0m+0DI' R = @B DI' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*(D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<* D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*D' =%(D6' =%(DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*DI' =4@BB5BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?RCBBCB#ppt_wB*Y3>B ppt_w<*D' =+4 8?RCBBCB#ppt_hB*Y3>B ppt_h<*D' =-g6B fade*<3<*D' =%(D=' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(DJ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*++0+0 ++0+0 ++0+0 +&   0 um&C(  2  NCo?"0@NNN?NTQ w 2  No?"0@NNN?NTw 2 ! No?"0@NNN?NTw 2 " No?"0@NNN?NTP w B - HDo?"0@NNN?N6 J 6 B . HDo?"0@NNN?N6 J G B / HDo?"0@NNN?NG x G B 0 HDo?"0@NNN?N6 x G B 1 HDo?"0@NNN?N6 x B 2 HDo?"0@NNN?N6 J  3 Bi o?"0@NNN?N Tb  hH 2 4 Bp o?"0@NNN?N6   hH 2B 5@ <Do?"0@NNN?N:  1 6 6lu  o?"6@ NNN?N  , JL(r0) coloringb 2 1 7 6}  o?"6@ NNN?N , JL(s0) coloringb 2 B 8 <Do?"0@NNN?N    < <Ā     B >@ <Do?"0@NNN?NJ0 6 B ?@ HDo?"0@NNN?NJ0 B @ HDo?"0@NNN?N0 % B A HDo?"0@NNN?N0 6 B B HDo?"0@NNN?N0 B C@ HDo?"0@NNN?NJ0 % B =@ <DԔ?"0@NNN?NJ0 % ,$D 0(F -  #  x 2 $ 6 -` > 2 % 6p -p > 2 & 6c -  > B ' ND>?"0@NNN?N%  (F -  (   2 ) 6 -` > 2 * 6 -p > 2 + 6 c -  > B , ND>?"0@NNN?N%  F   " 9  "2 : 6  ]  >  ; B o?"0@NNN?N\  " Gv* 2H  0޽h ? zw33fYyyYQ___PPT101.!@_ +lD' I = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-6B'blinds(horizontal)*<3<*=++%  0 '((  r  S     2  NCo?"0@NNN?NTQ w 2  No?"0@NNN?NTw 2  No?"0@NNN?NTw 2  No?"0@NNN?NTP w B  HDo?"0@NNN?N6 J 6 B  HDo?"0@NNN?N6 J G B  HDo?"0@NNN?NG x G B  HDo?"0@NNN?N6 x G B  HDo?"0@NNN?N6 x B  HDo?"0@NNN?N6 J   B o?"0@NNN?N Tb  hH 2  B o?"0@NNN?N6   hH 2B @ <Do?"0@NNN?N:  1  6X  o?"6@ NNN?N  , JL(r0) coloringb 2 1  6  o?"6@ NNN?N , JL(s0) coloringb 2 B  <Do?"0@NNN?N   B !@ <Do?"0@NNN?NJ0 6 B "@ HDo?"0@NNN?NJ0 B # HDo?"0@NNN?N0 % B $ HDo?"0@NNN?N0 6 B % HDo?"0@NNN?N0 B &@ HDo?"0@NNN?NJ0 % B '@ <DԔ?"0@NNN?NJ0 % B ( HDԔ?"0@NNN?N0 6 ,$D 0(F -    x 2   6 -` > 2   6 -p > 2   6L c -  > B   ND>?"0@NNN?N%  (F -      2  6 -` > 2  6| -p > 2  6 c -  > B  ND>?"0@NNN?N%  F   "   "2  6  ]  >    B o?"0@NNN?N\  " Gv* 2H  0޽h ? zw33fYyy___PPT10."+5EDx' I = @B D3' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-6B'blinds(horizontal)*<3<*(Dk' =%(D' =%(D9' =4@BBBB%(D' =/u8CC*Q3> Br<*'D9' =4@BBBB%(D' =/u8CC*Q3> Br<*(D9' =4@BBBB%(D' =/u8CC*Q3> Br<*+H3  0 %% )+)%(  :  0   @Furthermore, the v-to-S0 edges are either all red or all green.~A  2  NCo?"0@NNN?NTQ w 2  No?"0@NNN?NTw 2  No?"0@NNN?NTw 2  No?"0@NNN?NTP w B  HDo?"0@NNN?N6 J 6 B  HDo?"0@NNN?N6 J G B  HDo?"0@NNN?NG x G B  HDo?"0@NNN?N6 x G B  HDo?"0@NNN?N6 x B  HDo?"0@NNN?N6 J   B o?"0@NNN?N Tb  hH 2  B o?"0@NNN?N6   hH 2B @ <Do?"0@NNN?N:  1  6  o?"6@ NNN?N  , JL(r0) coloringb 2 1  6  o?"6@ NNN?N , JL(s0) coloringb 2 B  <Do?"0@NNN?N   B "@ <Do?"0@NNN?NJ0 6 B #@ HDo?"0@NNN?NJ0 B $ HDo?"0@NNN?N0 % B % HDo?"0@NNN?N0 6 B & HDo?"0@NNN?N0 B '@ HDo?"0@NNN?NJ0 % B (@ <DԔ?"0@NNN?NJ0 % B ) HDԔ?"0@NNN?N0 ,$D 0B * HDԔ?"0@NNN?N0 % ,$@ 0 +  0e0e    B>C^DEFA Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||> >>)5^@  "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab% & ,$D 0(F -     x 2   6X( -` > 2   60, -p > 2   6<0 c -  > B   ND>?"0@NNN?N%  (F -     2  64 -` > 2  67 -p > 2  6(< c -  > B  ND>?"0@NNN?N%  F   "   "2   6 @  ]  >  ! B4C o?"0@NNN?N\  " Gv* 2H  0޽h ? zw33fYyyW O ___PPT10/ ."Pl!+TD ' I = @B D ' = @BA?%,( < +O%,( < +DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-6B'blinds(horizontal)*<3<**DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-6B'blinds(horizontal)*<3<*)D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(Dk' =%(D' =%(D9' =4@BBBB%(D' =/u8CC*Q3> Br<**D9' =4@BBBB%(D' =/u8CC*Q3> Br<*)D9' =4@BBBB%(D' =/u8CC*Q3> Br<*++5o  0 44016=4(    S HL qT<$D  0   2  NCo?"0@NNN?N 7 2  No?"0@NNN?N( 2  No?"0@NNN?N( 2  No?"0@NNN?N 6 B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q   BP o?"0@NNN?N h  hH 2  BV o?"0@NNN?N   hH 2B @ <Do?"0@NNN?N :[1  6]  o?"6@ NNN?NZua  JL(r0) coloringb 2 1  6b  o?"6@ NNN?NZ, JL(s0) coloringb 2 B  <Do?"0@NNN?N &B !@ <Do?"0@NNN?N` B "@ HDo?"0@NNN?N` 9 B # HDo?"0@NNN?N` p B $ HDo?"0@NNN?N` p B % HDo?"0@NNN?N` p B &@ HDo?"0@NNN?N_ B '@ <DԔ?"0@NNN?N` B ) HDԔ?"0@NNN?N` p ,$@ 0B * HDԔ?"0@NNN?N` p ,$D 0B + HDԔ?"0@NNN?N` p ,$@ 0B -@ HDԔ?"0@NNN?N` 9 ,$D 0B .@ HDCԔ?"0@NNN?N` ,$D 0 0  0e0e    BCDEFA Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q-[q@   "0e@      @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab  ,$D 02 1 B21jJ?"0@NNN?N ,$@ 0 3 NXo o?"0@NNN?N/,$ 0 R=R0*"{v} (22 4 <jJ?"0@NNN?NP ,$D 0b 5 Ny o?"0@NNN?N) ,$ 0 xS=S0X(2 6  0e0e    BCDEF Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q-[q@   "0e@      @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab 9 ,$D 0(F -    J 2   6؁ -` > 2   6 -p > 2   6T c -  > B   ND>?"0@NNN?N%  (F -     CJ 2  6P -` > 2  64 -p > 2  6 c -  > B  ND>?"0@NNN?N%  F   "  P p2  6  ]  >    Bԛ o?"0@NNN?N\  " Gv* 2H  0޽h ? zw33fYyy0:(:___PPT10:."zg+sUDl9' I = @B D'9' = @BA?%,( < +O%,( < +D%' =%(D' =%(Dw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-6B'blinds(horizontal)*<3<*+Dw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-6B'blinds(horizontal)*<3<*)Dw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-6B'blinds(horizontal)*<3<**D' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%L%(D' =-}6B!strips(upRight)*<3<*%LD' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-6B'blinds(horizontal)*<3<*-D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(Dk' =%(D' =%(D9' =4@BBBB%(D' =/u8CC*Q3> Br<*-D9' =4@BBBB%(D' =/u8CC*Q3> Br<*'D9' =4@BBBB%(D' =/u8CC*Q3> Br<*6D ' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*-D' =1:Bhidden*o3>+B#style.visibility<*-%(D)' =4@BB BB%(D' =-g6B fade*<3<*6D' =1:Bhidden*o3>+B#style.visibility<*6%(D' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-}6B!strips(upRight)*<3<*LD' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-6B'blinds(horizontal)*<3<*.D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(Dk' =%(D' =%(D9' =4@BBBB%(D' =/u8CC*Q3> Br<*0D9' =4@BBBB%(D' =/u8CC*Q3> Br<*.D9' =4@BBBB%(D' =/u8CC*Q3> Br<*'D ' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*.D' =1:Bhidden*o3>+B#style.visibility<*.%(D)' =4@BB BB%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-}6B!strips(upRight)*<3<*D ' =%(Dn ' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-u6Bdiamond(in)*<3<*1DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-u6Bdiamond(in)*<3<*3DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-u6Bdiamond(in)*<3<*5D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-u6Bdiamond(in)*<3<*4+p+0+30 ++0+50 +h  0 00@28n0(  2  0x  qT,$D  0 Case 2 All v-to-S0 edges are red. Then all v-to-R0 edges are red, too. Now let R={v}, S=R0*"S0. Thus we have done.t    2  NCo?"0@NNN?N 7 2  No?"0@NNN?N( 2  No?"0@NNN?N( 2  No?"0@NNN?N 6 B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q B  HDo?"0@NNN?N q   B o?"0@NNN?N h  hH 2  B o?"0@NNN?N   hH 2B @ <Do?"0@NNN?N :[1  6  o?"6@ NNN?NZua  JL(r0) coloringb 2 1  6  o?"6@ NNN?NZ, JL(s0) coloringb 2 B  <Do?"0@NNN?N &B "@ <Do?"0@NNN?N` B #@ HDo?"0@NNN?N` 9 B $ HDo?"0@NNN?N` p B % HDo?"0@NNN?N` p B & HDo?"0@NNN?N` p B '@ HDo?"0@NNN?N_ B (@ <DԔ?"0@NNN?N` B ) HDԔ?"0@NNN?N` p ,$@ 0B * HDԔ?"0@NNN?N` p ,$D 0B + HDԔ?"0@NNN?N` p ,$@ 0B /@ HDԔ?"0@NNN?N` 9 ,$D 0 0  0e0e    BCDEFA Ԕ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||.\@  "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab 9 ,$D 0B 1@ HDԔ?"0@NNN?N` ,$D  0B 2@ HDԔ?"0@NNN?N` ,$D 0B 3@ HDԔ?"0@NNN?N` 9 ,$@ 02 4 <jJ?"0@NNN?NC,$@ 0 6 ND o?"0@NNN?Np ,$ 0 S=R0*"S0(22 7 BjJ?"0@NNN?N ,$D 0o 8 N o?"0@NNN?N S,$ 0 R={v}d(2(F -     J 2   6 -` > 2   6 -p > 2   6 c -  > B   ND>?"0@NNN?N%  (F -    CJ 2  6 -` > 2  6 -p > 2  6(# c -  > B  ND>?"0@NNN?N%  F   "  P p2   6\'  ]  >  ! B* o?"0@NNN?N\  " Gv* 2H  0޽h ? zw33fYyy77___PPT10`7. " 7+nqD6' I = @B D6' = @BA?%,( < +O%,( < +D' =%(D=' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-6B'blinds(horizontal)*<3<*+DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-6B'blinds(horizontal)*<3<*)DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-6B'blinds(horizontal)*<3<**D' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#I%(D' =-}6B!strips(upRight)*<3<*#ID' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-6B'blinds(horizontal)*<3<*/D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(Dk' =%(D' =%(D9' =4@BBBB%(D' =/u8CC*Q3> Br<*(D9' =4@BBBB%(D' =/u8CC*Q3> Br<*/D9' =4@BBBB%(D' =/u8CC*Q3> Br<*0D ' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*/D' =1:Bhidden*o3>+B#style.visibility<*/%(D)' =4@BB BB%(D' =-g6B fade*<3<*0D' =1:Bhidden*o3>+B#style.visibility<*0%(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-6B'blinds(horizontal)*<3<*1Dk' =%(D' =%(D9' =4@BBBB%(D' =/u8CC*Q3> Br<*)D9' =4@BBBB%(D' =/u8CC*Q3> Br<*1D9' =4@BBBB%(D' =/u8CC*Q3> Br<*D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*1D' =1:Bhidden*o3>+B#style.visibility<*1%(DF' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-6B'blinds(horizontal)*<3<*3DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-6B'blinds(horizontal)*<3<*2D' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*It%(D' =-}6B!strips(upRight)*<3<*ItD ' =%(Dn ' =%(D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-u6Bdiamond(in)*<3<*4DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =-u6Bdiamond(in)*<3<*6DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D' =-u6Bdiamond(in)*<3<*8D9' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*7%(D' =-u6Bdiamond(in)*<3<*7+p+0+60 ++0+80 +40  0 K/C/GZ.(  r  S ? Z   ).8  ZB HB BDo?"0@NNN?NP B I BDo?"0@NNN?N] PB KB BDo?"0@NNN?N  B L BDo?"0@NNN?N: B MB BDo?"0@NNN?Ngc  B N BDo?"0@NNN?Nc = B OB BDo?"0@NNN?NV =B P BDo?"0@NNN?NjV B QB BDo?"0@NNN?N3 Cp B R BDo?"0@NNN?NpC6 B TB BDo?"0@NNN?NC6 6 B U BDo?"0@NNN?Nc6  B VB BDo?"0@NNN?N V B W BDo?"0@NNN?N) B XB BDo?"0@NNN?NF) VB Y BDo?"0@NNN?NV I"@  J@ =  += 2  Ho?"0@NNN?N M   HB o?"0@NNN?N=  ;9(2@ wM %  ,`2   Ho?"0@NNN?NJ%    H0G o?"0@NNN?NwM   ;4(2@   %  - `2   Ho?"0@NNN?N  %    HK o?"0@NNN?N    ;5(2@  gF  3S  2   Ho?"0@NNN?Nd F   HO o?"0@NNN?N g3  ;1(2@ w M F  2S k 2  Ho?"0@NNN?NJ F   HxS o?"0@NNN?Nw M 3  ;3(2@  F  1 S p 2  Ho?"0@NNN?N F   HX o?"0@NNN?N 3  ;1(2@ *  4*s  2  Ho?"0@NNN?N :  H,V o?"0@NNN?N*   ;1(2@  n  6t 2  Ho?"0@NNN?Nk   H_ o?"0@NNN?N n  ;2(2@  V  /Ft  2  Ho?"0@NNN?NS    Hc o?"0@NNN?N V  ;2(2@  5g =2  Ho?"0@NNN?N  Hh o?"0@NNN?N ;1(2@ J  7 2  Ho?"0@NNN?NZ  Hm o?"0@NNN?NJ  ;1(2@ & c  .S  2 ) Ho?"0@NNN?N& c  * Hp o?"0@NNN?N& P  ;4(2 N  V  9  s  2 : Ho?"0@NNN?NS   ; H\u o?"0@NNN?N V  ;2(2 N  < ? 2 = Ho?"0@NNN?N > Hy o?"0@NNN?N ;1(2 N J  ?  2 @ Ho?"0@NNN?NZ A H$~ o?"0@NNN?NJ  ;1(2 N  B   2 C Ho?"0@NNN?N D H o?"0@NNN?N ;1(2 N J  E  2 F Ho?"0@NNN?NZ G H o?"0@NNN?NJ  ;1(2H  0޽h ? zw33fYyy___PPT10i.K" n+D=' I = @B +  0 WWuGW(  B  @ NDo?"0@NNN?N= B  NDo?"0@NNN?N" 9 B  @ NDo?"0@NNN?NB  NDo?"0@NNN?NV7B  @ NDo?"0@NNN?NoVlB  NDo?"0@NNN?NoV^PB @ NDo?"0@NNN?NV^B  NDo?"0@NNN?N B @ NDo?"0@NNN?N| B  NDo?"0@NNN?N B @ NDo?"0@NNN?NS^ lB  NDo?"0@NNN?NS" lB @ NDo?"0@NNN?N1  B  NDo?"0@NNN?N9 fB @ NDff3o?"0@NNN?NB  NDff3o?"0@NNN?N1^S#L  # wgQN =   = 2  Ho?"0@NNN?N M   Hx o?"0@NNN?N=  99(2N wM \   `2  Ho?"0@NNN?NJ%   H o?"0@NNN?NwM \  94(2N   \     `2 ! Ho?"0@NNN?N  %  " Hp o?"0@NNN?N  \  95(2N  g}  # S  2 $ Ho?"0@NNN?Nd F  % HԨ o?"0@NNN?N g}  91(2N w M }  & S k 2 ' Ho?"0@NNN?NJ F  ( HP o?"0@NNN?Nw M }  93(2N  }  )  S p 2 * Ho?"0@NNN?N F  + Hذ o?"0@NNN?N }  91(2N * R , *s  2 - Ho?"0@NNN?N : . H o?"0@NNN?N* R 91(2N  n R / t 2 0 Ho?"0@NNN?Nk  1 Hh o?"0@NNN?N n R 92(2N  V$ 2 Ft  2 3 Ho?"0@NNN?NS   4 Hм o?"0@NNN?N V$ 92(2N  5 g =2 6 Ho?"0@NNN?N 7 H  o?"0@NNN?N 91(2N J  8  2 9 Ho?"0@NNN?NZ : Ht o?"0@NNN?NJ  91(2N &   ; S  2 < Ho?"0@NNN?N& c  = H o?"0@NNN?N&   94(2N  V& >  s  2 ? Ho?"0@NNN?NS   @ HT o?"0@NNN?N V& 92(2N  A ? 2 B Ho?"0@NNN?N C H o?"0@NNN?N 91(2N J  D  2 E Ho?"0@NNN?NZ F H4 o?"0@NNN?NJ  91(2N  G   2 H Ho?"0@NNN?N I H o?"0@NNN?N 91(2N J  J  2 K Ho?"0@NNN?NZ L H o?"0@NNN?NJ  91(2B X@ HDo?"0@NNN?N ,$D  0B Y HDo?"0@NNN?N J ,$D  0B Z HDo?"0@NNN?NJ ,$@  0B [@ HDo?"0@NNN?Nj,$D  0B \ HDo?"0@NNN?N j,$@  0B ] HDo?"0@NNN?N Jj,$@  0B ^ HDo?"0@NNN?NP pS ,$D  0B _@ HDff3o?"0@NNN?N C ,$D   0B ` HDo?"0@NNN?NP C ,$D   0B a HDo?"0@NNN?NS p ,$@   0B b@ HDo?"0@NNN?NS Cp ,$@   0B c HDo?"0@NNN?NP  ,$@   0B d@ HDo?"0@NNN?N P ,$D  0B e HDo?"0@NNN?NS p,$@  0B f HDo?"0@NNN?N ,$@  0B g HDo?"0@NNN?N C,$@  0B i HDo?"0@NNN?NP ,$@  0B j HDo?"0@NNN?NCS ,$@  0B k HDo?"0@NNN?N ,$@  0B l HDo?"0@NNN?NCC ,$@  0B m HDo?"0@NNN?N ,$@  02 n <jJ?"0@NNN?N  ,$@  02 p <jJ?"0@NNN?N2^ ,$D  0B q HDo?"0@NNN?NP ,$@  0B r HDo?"0@NNN?NS p ,$@  0B s HDo?"0@NNN?N J ,$@  0B t HDo?"0@NNN?N JC ,$@  0B u HDo?"0@NNN?N  ,$@  0B v HDo?"0@NNN?NJP ,$@  0B w HDo?"0@NNN?NS J ,$@  0B x HDo?"0@NNN?N J ,$@   0B y HDo?"0@NNN?N J ,$@"  0B z HDo?"0@NNN?N wC ,$@!  0B { HDo?"0@NNN?NjP ,$D'  0B | HDo?"0@NNN?NS jp,$@&  0B } HDo?"0@NNN?N j,$@%  0B ~ HDo?"0@NNN?N =C,$@$  0B  HDo?"0@NNN?Nj ,$@#  08  , W ,2 M 6 jC > 2 N 6X  J  > 2 O 6  x  > 2 P 6< = > 2 Q 6<   , > 2 R 6L   p  > 2 S 6    > 2 T 6 C%   > 2 U 6 # ~  > H  0޽h ? zw33fYyy\[___PPT10[.N"w+lD[' I = @B Dg[' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-m6Bbox(in)*<3<*XD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Z%(D' =-m6Bbox(in)*<3<*ZD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Y%(D' =-m6Bbox(in)*<3<*YDS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*]%(D' =-m6Bbox(in)*<3<*]D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*\%(D' =-m6Bbox(in)*<3<*\D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D' =-m6Bbox(in)*<3<*[D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*^%(D' =-m6Bbox(in)*<3<*^D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*_%(D' =-m6Bbox(in)*<3<*_D ' =%(D4 ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a%(D' =-m6Bbox(in)*<3<*aD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c%(D' =-m6Bbox(in)*<3<*cD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D' =-m6Bbox(in)*<3<*bD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*`%(D' =-m6Bbox(in)*<3<*`D ' =%(D4 ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D' =-m6Bbox(in)*<3<*gD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*f%(D' =-m6Bbox(in)*<3<*fD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =-m6Bbox(in)*<3<*eD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D' =-m6Bbox(in)*<3<*dD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*n%(D' =-m6Bbox(in)*<3<*nD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(D' =-m6Bbox(in)*<3<*pD-' =%(D,' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*i%(D' =-m6Bbox(in)*<3<*iD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*j%(D' =-m6Bbox(in)*<3<*jD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*q%(D' =-m6Bbox(in)*<3<*qD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*k%(D' =-m6Bbox(in)*<3<*kD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(D' =-m6Bbox(in)*<3<*lD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' =-m6Bbox(in)*<3<*mD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*r%(D' =-m6Bbox(in)*<3<*rD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*s%(D' =-m6Bbox(in)*<3<*sD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*u%(D' =-m6Bbox(in)*<3<*uD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*v%(D' =-m6Bbox(in)*<3<*vD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*w%(D' =-m6Bbox(in)*<3<*wD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x%(D' =-m6Bbox(in)*<3<*xD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*z%(D' =-m6Bbox(in)*<3<*zD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*y%(D' =-m6Bbox(in)*<3<*yD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*~%(D' =-m6Bbox(in)*<3<*~D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*}%(D' =-m6Bbox(in)*<3<*}D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*|%(D' =-m6Bbox(in)*<3<*|D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*{%(D' =-m6Bbox(in)*<3<*{+}  0 $(  r  S bj8<:  b r  S Db<< b H  0޽h ? zw33fYyy___PPT10i.J"0i`+D=' I = @B +,   0 n(  r  S L j       <M   3<$ 0   H  0޽h ? zw33fYyy___PPT10.P";+MDj' I = @B D%' = @BA?%,( < +O%,( < +D\' =%(D' =%(DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*DR' =A@BB BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B%slide(fromBottom)*<3<*+8+0+0 +*   0  V(    Bp_  ,      6`   ,$ 0 Now, let tn be the number of essentially different JL(n) colorings. So tn is also the number of full binary trees with n leafs, if n>2.  - > (@H  0޽h ? zw33fYyy  ___PPT10.Q"3+ D'  |= @B D;' = @BA?%,( < +O%,( < +Dr' =%(D' =%(D' =A@BB:BB0B%()?D' =1:Bvisible*o3>+B#style.visibility<* %(D' =+4 8?dCB#ppt_w*2.5BCB#ppt_wB*Y3>B ppt_w<* D' =+4 8?fCB#ppt_h*0.01BCB#ppt_hB*Y3>B ppt_h<* D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<* D' =+4 8?`CB#ppt_h+1BCB#ppt_yB*Y3>B ppt_y<* D' =-g6B fade*<3<* +8+0+ 0 +%  0 <4(  x  c $4| $ v    0 ZA  ? ?j C   f  s *A ? ? wm  H  0޽h ? zw33fYyy___PPT10i.&y+D=' I = @B +C  0 -3%35oP2(  r  S D a    8 W    $  N o?"0@NNN?NW   ft4 2F(282  Bo?"0@NNN?NK Z  l DC  UCD ,$D 0B B HDo?"0@NNN?N6 B   HDo?"0@NNN?Ne6 B  B HDo?"0@NNN?NT I) B   HDo?"0@NNN?Nh G B B HDo?"0@NNN?N ) B  HDo?"0@NNN?N z ) T =  # qC\ 2  Ho?"0@NNN?N M   H, o?"0@NNN?N=  ;8(2T wM %  # r ]F 2  Ho?"0@NNN?NJ%   H o?"0@NNN?NwM   ;4(2T   %  #  F 2   Ho?"0@NNN?N  %  ! Hؓ o?"0@NNN?N    ;4(22 M TjJ?"0@NNN?N 2 N TjJ?"0@NNN?NDt WAl  ) S  T % ,$D 0B O TDo?"0@NNN?NG9 9  P N o?"0@NNN?N ) S  C different (2  l m' W'% ,$@ 02 Q TjJ?"0@NNN?Nb m' R  0e0e    BCDEFA jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||e"/EkS3@  "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab2~ { z DC  X pm ,$D 0B YB BDo?"0@NNN?N6 B Z BDo?"0@NNN?Ne6 B [B BDo?"0@NNN?NT I) B \ BDo?"0@NNN?Nh G B ]B BDo?"0@NNN?N ) B ^ BDo?"0@NNN?N z ) T =  _# qC\ 2 ` Ho?"0@NNN?N M  a Hğ o?"0@NNN?N=  ;8(2T wM %  b# r ]F 2 c Ho?"0@NNN?NJ%  d HT o?"0@NNN?NwM   ;4(2T   %  e#  F 2 f Ho?"0@NNN?N  %  g H o?"0@NNN?N    ;4(22 h TjJ?"0@NNN?N 2 i TjJ?"0@NNN?NDt WLl 3 ) S  m) 3 S ,$D 0B k ZDo?"0@NNN?Nq9 9  l T o?"0@NNN?N3 ) S  Bthe same (2 l  oPp,$@ 02 V TjJ?"0@NNN?N n  0e0e    BCDEFA jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||<5j<x@  "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abnH  0޽h ? zw33fYyyvn___PPT10N.&XOz+(xD"' I = @B D' = @BA?%,( < +O%,( < +D' =%(D/' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*W%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D' =-6B+checkerboard(across)*<3<*UD' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-6B'blinds(horizontal)*<3<*TD' =%(D/' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*o%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =-6B+checkerboard(across)*<3<*XD' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' =-6B'blinds(horizontal)*<3<*m+I  0 <<0S x<(      B  w8<G       <p  Fj   W/l wg/  t ,$D 0B  B HDo?"0@NNN?N= B   HDo?"0@NNN?N" 9 B  B HDo?"0@NNN?NB   HDo?"0@NNN?NV7B  B HDo?"0@NNN?NoVlB   HDo?"0@NNN?NVo^PB  B HDo?"0@NNN?NV^B   HDo?"0@NNN?N B  B HDo?"0@NNN?N|  B   HDo?"0@NNN?N B  B HDo?"0@NNN?N^ SlB   HDo?"0@NNN?N"S lB  B HDo?"0@NNN?N1  B   HDo?"0@NNN?N9 fB  B HDff3o?"0@NNN?NB   HDff3o?"0@NNN?N1^[#T   # wg/N =    = 2   Ho?"0@NNN?N M    H o?"0@NNN?N=  99(2N wM %    `2   Ho?"0@NNN?NJ%    H o?"0@NNN?NwM "  94(2N   %     `2   Ho?"0@NNN?N  %    HD o?"0@NNN?N  "  95(2N  gF    S  2   Ho?"0@NNN?Nd F    H o?"0@NNN?N gD  91(2N w M F    S k 2   Ho?"0@NNN?NJ F    Hx o?"0@NNN?Nw M D  93(2N  F     S p 2   Ho?"0@NNN?N F    H  o?"0@NNN?N D  91(2N *    *s  2   Ho?"0@NNN?N :   H8 o?"0@NNN?N*  91(2N  n    t 2   Ho?"0@NNN?Nk    H o?"0@NNN?N n  92(2N  V    Ft  2   Ho?"0@NNN?NS     H o?"0@NNN?N V  92(2N    g =2   Ho?"0@NNN?N   H o?"0@NNN?N 91(2N J     2   Ho?"0@NNN?NZ   H o?"0@NNN?NJ  91(2N & c    S  2   Ho?"0@NNN?N& c    HD o?"0@NNN?N& a  94(2N  V     s  2   Ho?"0@NNN?NS     H o?"0@NNN?N V  92(2N    ? 2   Ho?"0@NNN?N   H o?"0@NNN?N 91(2N J     2   Ho?"0@NNN?NZ   H o?"0@NNN?NJ  91(2N      2   Ho?"0@NNN?N   H o?"0@NNN?N 91(2N J     2   Ho?"0@NNN?NZ   HH o?"0@NNN?NJ  91(2   0    ;,$ 0 M[The frequencies of the 8 colors are 20, 4, 4, 3, 2, 1, 1, 1. (in non-increasing order) \$ }l '   # ,$D 0   ND' o?"0@NNN?N -  ;1(2   N+ o?"0@NNN?N   ;2(2   N* o?"0@NNN?N'  ;3(2   NX2 o?"0@NNN?Nj <20(2   Nh6 o?"0@NNN?N ~   ;4(2   N : o?"0@NNN?N~ z  ;4(2   N= o?"0@NNN?N   ;1(2   NTA o?"0@NNN?NK   ;1(2=   ND o?"0@NNN?N@,$ 0 S . (2H   0޽h ? zw33fYyy[ S ___PPT103 .&xH+$\D ' I = @B DR ' = @BA?%,( < +O%,( < +D[' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B+checkerboard(across)*<3<* DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B+checkerboard(across)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(D:' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-i6B wedge*<3<* +p+0+ 0 ++0+ 0 +  0 .&@(    c $] m    H  0޽h ? zw33fYyy___PPT10i.J"(t+D=' I = @B +  0 0(P(    6e  8< G Equalized"    0Pj  #FjS  sAThe greatest color frequency is no more than the least plus one.BBH  0޽h ? zw33fYyy___PPT10i.&˒[+D=' I = @B +  0 `0(    S %c<$ 0   R  6p  ZC Theorem 3.1 Suppose n>1 and Kn is edge colored so that there are no rainbow K3 s. Then for at least one color, the subgraph of Kn induced by the edges of that color is connected and spanning in Kn.x  - 1 ? >UEH  0޽h ? zw33fYyy___PPT10 .&+GD ' I = @B D= ' = @BA?%,( < +O%,( < +Dt ' =%(D ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-6B'blinds(horizontal)*<3<*/DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*/`%(D' =-6B'blinds(horizontal)*<3<*/`DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*`n%(D' =-6B'blinds(horizontal)*<3<*`n+8+0+0 +&  0 ;3p(  2  N33Ԕ?"0@NNN?N2 2  N33jJ?"0@NNN?N2 ,$D 0r  S  :   @  Nh o?"0@NNN?N,$@  0 VIf any edge incident to v is green, then we have done. So suppose that green does NOT appear on a v-to-(Kn-v) edge. 8 vK0vK? vK(2' $e  B  HDo?"0@NNN?NAB(   B4 o?"0@NNN?N 2 b  Gv* 2B   HDo?"0@NNN?Nm B B   HDo?"0@NNN?N B3 B   HDo?"0@NNN?N B( 8  B o?"0@NNN?N Y Kn-vP 2B  HDԔ?"0@NNN?NBC) ,$D 02  6̮  o5  > 2  6ز Uo > 2  60 @ U  > 2  6  mi  > 2  6,  U  > H  0޽h ? zw33fYyy___PPT10.&PD@+܎D' I = @B Dm' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D:' =%(D' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-}6B!strips(upRight)*<3<*%DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%8%(D' =-}6B!strips(upRight)*<3<*%8D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(D)' =4@BB BB%(D' =-g6B fade*<3<*D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D+' =%(DA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9U%(D' =-}6B!strips(upRight)*<3<*9UDA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ur%(D' =-}6B!strips(upRight)*<3<*UrDA' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*rx%(D' =-}6B!strips(upRight)*<3<*rx+0A  0 SK (   B   HDo?"0@NNN?N6R r   S ķ       H o?"0@NNN?NS&  Gv* 2%   B o?"0@NNN?Na  ,$ 0 Gw* 2.   # 0e0e    BCnDE4FA Ao 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E|| a=z kKn @    "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab ,$D 0B   HDo?"0@NNN?N6v@ B   HDo?"0@NNN?N@ 6~ B   HDo?"0@NNN?N@ 6 B   HDԔ?"0@NNN?N6v@ ,$D   0B   HDԔ?"0@NNN?N@ 6~ ,$D   0B   HDԔ?"0@NNN?N@ 6 ,$D  02   6\ I > %   Bx o?"0@NNN?N,$ 0 Gu* 22   6 Y a  > 2   6  IU  >    N o?"0@NNN?NP l,$ 0 For some wV(Kn)-{u, v}, $ a green path from u to w. Since no v-to-(Kn-v) edge is green. Then all edges incident to v are red. vK    , - 92   NjJ?"0@NNN?N} 8 B   HDԔ?"0@NNN?N7S ,$D 08   B  o?"0@NNN?N M 8 Kn-vP 22   6` % c  > 2   68 _I > H   0޽h ? zw33fYyy$}$___PPT10]$.&z+>D#' I = @B DD#' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* DD' =%(D' =%(DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-u6Bdiamond(in)*<3<* DF' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* 7%(D' =-u6Bdiamond(in)*<3<* 7DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* D ' =%(D ' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* 7U%(D' =-6B'blinds(horizontal)*<3<* 7UDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* U\%(D' =-6B'blinds(horizontal)*<3<* U\DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* \z%(D' =-6B'blinds(horizontal)*<3<* \zDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* z%(D' =-6B'blinds(horizontal)*<3<* zD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* ++0+ 0 ++0+ 0 ++0+ 0 +P&  0 {s$ (  $ $ c $KcP<$ 0    $ 60   {Corollary 3.2 If n >1 and t > , then there is no equalized rainbow-K3-forbidding edge coloring of Kn with t colors.|n 0  j $0 NA "? ?> s H "$D 0  f $ s *A x? ?j  xH $ 0޽h ? zw33fYyy}!u!___PPT10U!.&x+>)KD ' A = @B D ' = @BA?%,( < +O%,( < +D' =%(D' =%(Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*$Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*$Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$/%(D' =-g6B fade*<3<*$/D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$/D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*$/Dp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$/a%(D' =-g6B fade*<3<*$/aD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$/aD' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*$/aDp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$a%(D' =-g6B fade*<3<*$aD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$aD' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*$aDp' =A@BB*BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*$Dc' =4@BB*BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-g6B fade*<3<*$D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$D' =+4 8?bCB#ppt_y+.1BCB#ppt_yB*Y3>B ppt_y<*$+8+0+$0 +!  0 ,,(  , , S $S L<$D 0    , 6T  Y (Theorem 3.3 For n >1, there is an equalized rainbow-K3-forbidding edge coloring of Kn with t colors iff t { 1, 2, & , }x "   ,S ,  nAo?v}"6@ NNN?Nw ,  nAo?v}"6@ NNN?N=f , s *A z? ?J z , s *A ? ?- Xs 8 $D 0H , 0޽h ? zw33fYyy___PPT10.&e+?wD' = @B Du' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,D%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,DD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,DD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,Du%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,DuD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,DuD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,u%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,uD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,uD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*,D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*,+*  0 "0(  0 0 c $u ,<$@ 0    00 NA ;? ?`  ;    0 s *A 2? ? U 8 2$@ 0 0 s *A 4? ?W 8 4$D 0H 0 0޽h ? zw33fYyy2'*'___PPT10 '.&` +RD&' = @B D&' = @BA?%,( < +O%,( < +D' =%(DW' =%(D' =4@BB2BB%())?)D' =1:Bvisible*o3>+B#style.visibility<*0g%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*0gD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*0gD' =-g6B fade*<3<*0gD' =4@BB2BB%())?)D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*0D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*0D' =-g6B fade*<3<*0D' =4@BB2BB%())?)D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*0D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*0D' =-g6B fade*<3<*0D' =4@BB2BB%())?)D' =1:Bvisible*o3>+B#style.visibility<* 0%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<* 0D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<* 0D' =-g6B fade*<3<* 0D' =4@BB2BB%())?)D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*0D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*0D' =-g6B fade*<3<*0D' =%(D ' =%(Ds' =4@BB2BB%()?D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*0D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*0D' =-g6B fade*<3<*0Ds' =4@BB2BB%()?D' =1:Bvisible*o3>+B#style.visibility<*0(%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*0(D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*0(D' =-g6B fade*<3<*0(Ds' =4@BB2BB%()?D' =1:Bvisible*o3>+B#style.visibility<*0(0%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*0(0D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*0(0D' =-g6B fade*<3<*0(0+0  0  D(  D D S   E'+<$ 0   T D 6   How to partition {1, 2, & , n-1} into t sets C1, C2, & , Ct, s.t. ? o3<0f  D s *A =? ?@ =H D 0޽h ? zw33fYyy++___PPT10+.' j+[W_D"+'  = @B D*' = @BA?%,( < +O%,( < +Dr' =%(D' =%(D' =A@BB:BB0B%()?D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB#ppt_w*2.5BCB#ppt_wB*Y3>B ppt_w<*DD' =+4 8?fCB#ppt_h*0.01BCB#ppt_hB*Y3>B ppt_h<*DD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?`CB#ppt_h+1BCB#ppt_yB*Y3>B ppt_y<*DD' =-g6B fade*<3<*DD"' =%(DB"' =%(D' =A@BB:BB0B%()?D' =1:Bvisible*o3>+B#style.visibility<*D'%(D' =+4 8?dCB#ppt_w*2.5BCB#ppt_wB*Y3>B ppt_w<*D'D' =+4 8?fCB#ppt_h*0.01BCB#ppt_hB*Y3>B ppt_h<*D'D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D'D' =+4 8?`CB#ppt_h+1BCB#ppt_yB*Y3>B ppt_y<*D'D' =-g6B fade*<3<*D'D' =A@BB:BB0B%()?D' =1:Bvisible*o3>+B#style.visibility<*D'V%(D' =+4 8?dCB#ppt_w*2.5BCB#ppt_wB*Y3>B ppt_w<*D'VD' =+4 8?fCB#ppt_h*0.01BCB#ppt_hB*Y3>B ppt_h<*D'VD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D'VD' =+4 8?`CB#ppt_h+1BCB#ppt_yB*Y3>B ppt_y<*D'VD' =-g6B fade*<3<*D'VD' =A@BB:BB0B%()?D' =1:Bvisible*o3>+B#style.visibility<*DV%(D' =+4 8?dCB#ppt_w*2.5BCB#ppt_wB*Y3>B ppt_w<*DVD' =+4 8?fCB#ppt_h*0.01BCB#ppt_hB*Y3>B ppt_h<*DVD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DVD' =+4 8?`CB#ppt_h+1BCB#ppt_yB*Y3>B ppt_y<*DVD' =-g6B fade*<3<*DVD' =A@BB:BB0B%()?D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB#ppt_w*2.5BCB#ppt_wB*Y3>B ppt_w<*DD' =+4 8?fCB#ppt_h*0.01BCB#ppt_hB*Y3>B ppt_h<*DD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?`CB#ppt_h+1BCB#ppt_yB*Y3>B ppt_y<*DD' =-g6B fade*<3<*DD' =A@BB:BB0B%()?D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =+4 8?dCB#ppt_w*2.5BCB#ppt_wB*Y3>B ppt_w<*DD' =+4 8?fCB#ppt_h*0.01BCB#ppt_hB*Y3>B ppt_h<*DD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*DD' =+4 8?`CB#ppt_h+1BCB#ppt_yB*Y3>B ppt_y<*DD' =-g6B fade*<3<*D+8+0+D0 +m  0 @%WT(  TR 8    UT @   ET s 2 CT No?"0@NNN?N   DT N o?"0@NNN?N7   NC10(2(N   FT V + 2 GT No?"0@NNN?N   HT N*B ppt_xB ppt_y=B0BB aaApBBI۾B$l><*@TD' =%(D' =%(D8' =A@BBBB0B%()?)?D' =.G7 BBBBBYM 1.94444E-6 4.45087E-6 L -0.08056 0.29664 *3>*B ppt_xB ppt_y=@0BBAApBBP$B><*1TDq' =%(D' =%(D' =A@BBBB0B%()?)?D' =.7 BBBBBM 1.11111E-6 -6.35838E-7 C -0.1559 0.02474 -0.31163 0.04971 -0.41267 0.09734 C -0.51371 0.14497 -0.57291 0.25364 -0.60642 0.28531 *3>*B ppt_xB ppt_y=0BB aaA<*ATD' =%(D' =%(D6' =A@BBBB0B%()?)?D}' =.E7 BBBBBWM 1.11111E-6 4.45087E-6 L 0.01371 0.33364 *3>*B ppt_xB ppt_y=@0BBAApBB;BŴ*><*2TDs' =%(D' =%(D' =A@BBBB0B%()?)?D ' =.7 BBBBB M -5.27778E-6 -2.48555E-6 C -0.08942 0.02659 -0.17882 0.05318 -0.23803 0.09942 C -0.29723 0.14566 -0.33577 0.2474 -0.35556 0.27699 *3>*B ppt_xB ppt_y=0BB aaA<*BTD' =%(D' =%(D6' =A@BBBB0B%()?)?D}' =.E7 BBBBBWM 2.77778E-7 4.45087E-6 L 0.10851 0.33872 *3>*B ppt_xB ppt_y=@0BBAApBB]=B~O-><*3TD9' =%(D' =%(D' =A@BBBB0B%()?)?D' =.7 BBBBBM -4.16667E-6 -6.35838E-7 C 0.01667 0.07722 0.03334 0.15468 0.04757 0.16924 C 0.06181 0.18381 0.09931 0.10196 0.08559 0.0867 C 0.07188 0.07144 0.00139 0.0467 -0.03507 0.07815 C -0.07153 0.10959 -0.10243 0.19213 -0.13333 0.27491 *3>*B ppt_xB ppt_y=0BB aaaaA<*?TD' =%(Di' =%(D' =A@BBBB0B%()?)?DX' =. 7 BBBBBYM 2.77778E-7 -5.78035E-6 L 0.15764 0.32508 *3>*B ppt_xB ppt_y=0BBAA<*6TD ' =%(D ' =%(Di ' =A@BBBB0B%()?)?D' =.x7 BBBBBM 6.66667E-6 -9.24855E-6 C -0.00121 0.01248 -0.0026 0.02404 -0.00485 0.03606 C -0.00365 0.06913 -0.01128 0.07745 0.00782 0.08254 C 0.01598 0.08786 0.02449 0.09248 0.03334 0.09525 C 0.03438 0.09734 0.03629 0.09895 0.03647 0.1015 C 0.03786 0.1163 0.03438 0.13202 0.03803 0.14589 C 0.03907 0.15005 0.04445 0.1445 0.04758 0.14381 C 0.05817 0.1445 0.06945 0.14104 0.07935 0.14589 C 0.0823 0.14728 0.07848 0.15445 0.07779 0.15861 C 0.07657 0.16578 0.07275 0.16971 0.0698 0.17549 C 0.07135 0.17757 0.0724 0.1808 0.07449 0.18196 C 0.07848 0.1845 0.08732 0.18612 0.08732 0.18612 C 0.09445 0.19075 0.10053 0.19468 0.10313 0.20508 C 0.10261 0.20855 0.10296 0.21271 0.10157 0.21572 C 0.09966 0.22011 0.09358 0.22635 0.09358 0.22635 C 0.0901 0.24046 0.08803 0.25641 0.10157 0.26219 C 0.10452 0.27375 0.10296 0.26312 0.10001 0.27699 C 0.0981 0.28554 0.09722 0.29387 0.09515 0.30242 C 0.0941 0.30104 0.09289 0.29965 0.09202 0.29803 C 0.09081 0.29595 0.0889 0.29179 0.0889 0.29179 *3>*B ppt_xB ppt_y==0BB+fffffffffffffffffffA<*8TD' =%(DC' =%(D' =A@BBBB0B%()?)?D2' =.7 BBBBB3M 0 0 L 0.26771 0.32509 *3>*B ppt_xB ppt_y=0BBAA<*5TD#' =%(D,#' =%(D' =A@BBBB0B%()?)?Db' =.*7 BBBBB_M 2.22222E-6 -7.39884E-6 C 0.11163 0.05294 0.22326 0.10612 0.30312 0.11837 C 0.38299 0.13063 0.43628 0.0497 0.47934 0.07398 C 0.52239 0.09826 0.54219 0.18126 0.56198 0.26427 *3>*B ppt_xB ppt_y=0BB aaaA<*4TD ' =A@BBBB0B%()?)?D ' =. 7 BBBBBM -8.33333E-7 4.45087E-6 C 0.0099 0.01063 0.02083 0.01965 0.03073 0.02982 C 0.05573 0.05526 0.07934 0.0793 0.10868 0.09433 C 0.12431 0.10242 0.14219 0.09826 0.15886 0.09896 C 0.16997 0.10474 0.15868 0.09942 0.17986 0.10381 C 0.1934 0.10659 0.20695 0.10982 0.22031 0.11283 C 0.25104 0.1193 0.28056 0.12786 0.31129 0.13133 C 0.34896 0.15121 0.4158 0.14011 0.43906 0.14057 C 0.45972 0.14265 0.48056 0.14913 0.50104 0.14982 C 0.53924 0.15144 0.57778 0.15144 0.61597 0.15213 C 0.63386 0.16046 0.64983 0.17109 0.66788 0.17734 C 0.66962 0.1778 0.67118 0.17872 0.67274 0.17965 C 0.67483 0.1808 0.67691 0.18335 0.67917 0.1845 C 0.68281 0.18589 0.68681 0.18589 0.69045 0.18659 C 0.69497 0.1926 0.69601 0.19768 0.69861 0.20508 C 0.70139 0.21271 0.70573 0.21734 0.71007 0.22358 C 0.71163 0.22566 0.7132 0.2282 0.71476 0.23052 C 0.7158 0.23213 0.71823 0.23514 0.71823 0.23537 C 0.72118 0.24786 0.72274 0.25317 0.72951 0.26289 C 0.73038 0.26635 0.73472 0.28092 0.73594 0.28346 C 0.7382 0.28786 0.7441 0.29502 0.7441 0.29526 C 0.7474 0.30913 0.74688 0.31768 0.75695 0.32254 C 0.76076 0.32809 0.76337 0.32716 0.76684 0.33202 *3>*B ppt_xB ppt_y=j0BB1ffffffffffffffffffffffApBBDD>B)><*'TD' =A@BBBB0B%()?)?D ' =. 7 BBBBB M 1.38778E-17 4.45087E-6 C 0.01198 0.01271 0.02205 0.02705 0.03524 0.03815 C 0.04271 0.05387 0.03403 0.03861 0.04323 0.04739 C 0.04514 0.04924 0.04618 0.05225 0.04809 0.0541 C 0.05 0.05618 0.05243 0.05711 0.05451 0.05849 C 0.06354 0.0719 0.08073 0.08462 0.09306 0.09017 C 0.09722 0.09618 0.09653 0.09687 0.10417 0.09711 C 0.13958 0.09872 0.175 0.09872 0.21024 0.09942 C 0.21181 0.09988 0.2217 0.1015 0.22465 0.10381 C 0.23229 0.11052 0.2217 0.11213 0.2375 0.11745 C 0.24722 0.12092 0.25833 0.12393 0.26788 0.12878 C 0.275 0.13248 0.27951 0.13549 0.28715 0.1378 C 0.29896 0.14635 0.31233 0.14751 0.32413 0.15606 C 0.34774 0.15537 0.37118 0.15583 0.39479 0.15352 C 0.39809 0.15329 0.39965 0.14659 0.40278 0.14474 C 0.41424 0.1378 0.42569 0.13502 0.43802 0.13341 C 0.43958 0.13248 0.44132 0.13109 0.44306 0.13109 C 0.45226 0.13109 0.46111 0.13202 0.47014 0.13341 C 0.47951 0.13456 0.48576 0.14751 0.4941 0.15144 C 0.50139 0.16208 0.51545 0.16208 0.52483 0.16716 C 0.56094 0.16601 0.58438 0.17202 0.61458 0.15838 C 0.62066 0.15907 0.62656 0.15884 0.63212 0.16046 C 0.63368 0.16115 0.6342 0.16393 0.63576 0.16508 C 0.64184 0.16924 0.65139 0.17202 0.65816 0.1741 C 0.69566 0.17225 0.73299 0.17109 0.77031 0.17641 C 0.77847 0.18011 0.78455 0.19121 0.79149 0.19445 C 0.80781 0.20208 0.78351 0.19005 0.80087 0.20115 C 0.80399 0.20323 0.80764 0.20393 0.81059 0.20578 C 0.81267 0.20739 0.81493 0.20878 0.81701 0.2104 C 0.82708 0.22474 0.83819 0.24531 0.84271 0.2645 C 0.84375 0.26982 0.84809 0.27213 0.85069 0.27583 C 0.8592 0.28809 0.85521 0.28578 0.85885 0.29618 C 0.86372 0.30982 0.86823 0.32369 0.86823 0.33942 *3>*B ppt_xB ppt_y=~0BBEffffffffffffffffffffffffffffffffApBB8>B-><*TDC' =A@BBBB0B%()?)?D' =.R7 BBBBBwM 3.33333E-6 -2.48555E-6 C 0.00503 0.00485 0.00677 0.01017 0.01267 0.01271 C 0.02118 0.02404 0.03107 0.03329 0.04132 0.04023 C 0.05225 0.04763 0.04583 0.04508 0.05555 0.05295 C 0.06614 0.0615 0.08316 0.07006 0.09531 0.07399 C 0.10642 0.08185 0.11944 0.0867 0.13177 0.08878 C 0.14114 0.09341 0.15069 0.09734 0.16024 0.1015 C 0.21024 0.10081 0.30347 0.11745 0.35712 0.06982 C 0.36285 0.0585 0.35868 0.05318 0.35243 0.04439 C 0.33333 0.04717 0.33264 0.04347 0.32864 0.06566 C 0.32812 0.11769 0.32691 0.16994 0.32691 0.22196 C 0.32691 0.23815 0.33107 0.2511 0.32535 0.26636 *3>*B ppt_xB ppt_y=-0BBfffffffffffA<*7T++0+T0 ++0+'T0 ++0+1T0 ++0+2T0 ++0+3T0 ++0+4T0 ++0+5T0 ++0+6T0 ++0+7T0 ++0+8T0 ++0+?T0 ++0+@T0 ++0+AT0 ++0+BT0 +4  0 ..7;tO.(  tB t@ HDo?"0@NNN?N Cp,$D  0B t HDo?"0@NNN?Np s ,$D  0B t HDo?"0@NNN?N Cs ,$@  0B t@ HDo?"0@NNN?NCf ,$D  0B t HDo?"0@NNN?Np f ,$@  0B  t HDo?"0@NNN?Ns f ,$@  0B  t HDo?"0@NNN?N},$@  0B  t@ HDo?"0@NNN?Nc V ,$@  0B  t HDo?"0@NNN?N} ,$@   0B  t HDo?"0@NNN?Nc ,$@  0B t@ HDo?"0@NNN?NV ,$@  0B t HDo?"0@NNN?N}c ,$@  0B t@ HDo?"0@NNN?N}sf ,$@!  0B t HDo?"0@NNN?Nt ,$@  0B t HDo?"0@NNN?Nc s ,$@  0B t HDo?"0@NNN?NV s ,$D  0B t HDo?"0@NNN?N}C,$D%  0B t HDo?"0@NNN?NC,$D  0B t HDo?"0@NNN?NCc ,$D  0B t HDo?"0@NNN?NCV ,$@   0B t HDo?"0@NNN?NCs ,$D   0B t HDo?"0@NNN?N} p,$@$  0B t HDo?"0@NNN?N p,$@  0B t HDo?"0@NNN?Np c ,$@  0B t HDo?"0@NNN?Np V ,$@   0B t HDo?"0@NNN?Np s ,$@   0B  t HDo?"0@NNN?N} s ,$@#  0B !t HDo?"0@NNN?N Is ,$@  0B "t HDo?"0@NNN?Nc s ,$@  0B #t HDo?"0@NNN?Ns s ,$@   0B $t HDo?"0@NNN?Ns " V ,$@  0B %t HDo?"0@NNN?N}f ,$@"  0B &t HDo?"0@NNN?Nf ,$@  0B 't HDo?"0@NNN?Nc f ,$@  0B (t HDo?"0@NNN?NV f ,$@  0B )t HDo?"0@NNN?Nf s ,$@  0 4t <t;  "A <$& 0   B 6t@ HDo?"0@NNN?N Cp,$D9 0A 5t 0H=   m U Find an equalized rainbow-K3-forbidding edge coloring of K9 with 5 colors.pV  B 7t HDo?"0@NNN?Np s ,$@A 0B 8t HDo?"0@NNN?N Cs ,$DB 0B 9t@ HDo?"0@NNN?NCf ,$@J 0B :t HDo?"0@NNN?Np f ,$@K 0B ;t HDo?"0@NNN?Ns f ,$DL 0F  , *t P  2 +t 6L jC > 2 ,t 6O  J  > 2 -t 6S  x  > 2 .t 60W = > 2 /t 6\Z   , > 2 0t 6]  p  > 2 1t 6b    > 2 2t 6e C%   > 2 3t 6h # ~  > H t 0޽h ? zw33fYyy___PPT10.H-+ mD'  = @B DL' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tDS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-m6Bbox(in)*<3<* tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD ' =%(D4 ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)t%(D' =-m6Bbox(in)*<3<*)tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#t%(D' =-m6Bbox(in)*<3<*#tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD ' =%(Dm ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$t%(D' =-m6Bbox(in)*<3<*$tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(t%(D' =-m6Bbox(in)*<3<*(tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD ' =%(D ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-m6Bbox(in)*<3<* tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'t%(D' =-m6Bbox(in)*<3<*'tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"t%(D' =-m6Bbox(in)*<3<*"tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD7' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-m6Bbox(in)*<3<* tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&t%(D' =-m6Bbox(in)*<3<*&tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!t%(D' =-m6Bbox(in)*<3<*!tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tDp' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-m6Bbox(in)*<3<* tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-m6Bbox(in)*<3<* tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%t%(D' =-m6Bbox(in)*<3<*%tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* t%(D' =-m6Bbox(in)*<3<* tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(D' =-m6Bbox(in)*<3<*tD' =%(DM' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4t%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4t*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4t*<%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4t<Q%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4tQX%(D' =%(D' =%(D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*!tD' =0l9 BBBB*<3<*!t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*&tD' =0l9 BBBB*<3<*&t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* tD' =0l9 BBBB*<3<* t)?D' =%(D' =%(D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* tD' =0l9 BBBB*<3<* t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*'tD' =0l9 BBBB*<3<*'t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*"tD' =0l9 BBBB*<3<*"t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D' =%(Dw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*6t%(D' =-6B'blinds(horizontal)*<3<*6tD@' =%(D' =%(D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*(tD' =0l9 BBBB*<3<*(t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*$tD' =0l9 BBBB*<3<*$t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?DN' =%(Dw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*7t%(D' =-6B'blinds(horizontal)*<3<*7tDw' =4@BBB B%()))D' =1:Bvisible*o3>+B#style.visibility<*8t%(D' =-6B'blinds(horizontal)*<3<*8tD/' =%(D' =%(D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*)tD' =0l9 BBBB*<3<*)t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*#tD' =0l9 BBBB*<3<*#t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* tD' =0l9 BBBB*<3<* t)?D=' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*9t%(D' =-6B'blinds(horizontal)*<3<*9tDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*:t%(D' =-6B'blinds(horizontal)*<3<*:tDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*;t%(D' =-6B'blinds(horizontal)*<3<*;tD' =%(D' =%(D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* tD' =0l9 BBBB*<3<* t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* tD' =0l9 BBBB*<3<* t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*%tD' =0l9 BBBB*<3<*%t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<* tD' =0l9 BBBB*<3<* t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?D&' =4@BBBB%(D' =?B70, 0; .2, .5; .8, .5; 1, 0-g6B fade*<3<*tD' =0l9 BBBB*<3<*t)?+8+0+4t0 +  0 .&|(  | | c $tf2a     H | 0޽h ? zw33fYyy___PPT10i.J"`5b+D=' I = @B +   0  x~(  xB x@ HDo?"0@NNN?N = ,$D 0B x HDo?"0@NNN?Ns * B x HDo?"0@NNN?N *=s r x S ز S    r x S   \   2 x <  =  > 2 x <p F W  > 2 x <|  j  > H x 0޽h ? zw33fYyy___PPT10n.R4'+z TDB' I = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*x%(+  0 (    s  0e0e #" 0e     f  s *A ~? ?4X z ~H  0޽h ? zw33fYyy___PPT10i.U4`[F+D=' I = @B +   0 X(  r  S        S  r<$ 0   f  s *A ? ?F]   s *A ? ?  $8 $D 0H  0޽h ? zw33fYyy___PPT10.V4P(+ǵ,D]' I = @B D' = @BA?%,( < +O%,( < +DO' =%(D' =%(Dp' =A@BBB B0B@%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*Dp' =A@BBB B0B@%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-6B'blinds(horizontal)*<3<*"Dp' =A@BBB B0B@%(D' =1:Bvisible*o3>+B#style.visibility<*#Y%(D' =-6B'blinds(horizontal)*<3<*#YDp' =A@BBB B0B@%(D' =1:Bvisible*o3>+B#style.visibility<*Yq%(D' =-6B'blinds(horizontal)*<3<*YqDp' =A@BBB B0B@%(D' =1:Bvisible*o3>+B#style.visibility<*q%(D' =-6B'blinds(horizontal)*<3<*qDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+8+0+0 +[K  0  77 Qm0 6(  mz t c  gI,$D 0T gV  d# tB eB BDo?"0@NNN?NV =B f BDo?"0@NNN?NjV T  g# g =2 h Ho?"0@NNN?N i H o?"0@NNN?N ;1(2T J  j#  2 k Ho?"0@NNN?NZ l HP o?"0@NNN?NJ  ;1(2 m B o?"0@NNN?N ;1(2mz t X %   ,$@ 0T gV  Y# tB ZB BDo?"0@NNN?NV =B [ BDo?"0@NNN?NjV T  \# g =2 ] Ho?"0@NNN?N ^ H o?"0@NNN?N ;1(2T J  _#  2 ` Ho?"0@NNN?NZ a H o?"0@NNN?NJ  ;1(2 b B@ o?"0@NNN?N ;1(2mz t M Z ,$@ 0T gV  N# tB OB BDo?"0@NNN?NV =B P BDo?"0@NNN?NjV T  Q# g =2 R Ho?"0@NNN?N S H  o?"0@NNN?N ;1(2T J  T#  2 U Ho?"0@NNN?NZ V H o?"0@NNN?NJ  ;1(2 W B o?"0@NNN?N ;1(2ql t Lt,$@ 0T gV  I# tB  B HDo?"0@NNN?NV =B   HDo?"0@NNN?NjV T  1# g =2 2 Ho?"0@NNN?N 3 Ht o?"0@NNN?N ;1(2T J  4#  2 5 Ho?"0@NNN?NZ 6 H o?"0@NNN?NJ  ;1(2 J H o?"0@NNN?N ;1(2r  S  !    B @ HDo?"0@NNN?NB  HDo?"0@NNN?N B  @ HDo?"0@NNN?Nm] B  HDo?"0@NNN?N SB @ HDo?"0@NNN?N&B  HDo?"0@NNN?N`% B @ HDo?"0@NNN?NS aF B  HDo?"0@NNN?NGF L =  #  \2  Ho?"0@NNN?N M   H4' o?"0@NNN?N=  ;9(2L wM %  #  2  Ho?"0@NNN?NJ%   H|- o?"0@NNN?NwM   ;2(2L   %  # \\ 2 2  Ho?"0@NNN?N  %   H/ o?"0@NNN?N    ;7(2L  F  %# } 2 & Ho?"0@NNN?N F  ' H4 o?"0@NNN?N 3  ;2(2L  V  .# g 2 / Ho?"0@NNN?NS   0 H8 o?"0@NNN?N V  ;3(2L & c  7# } 2 8 Ho?"0@NNN?N& c  9 HX= o?"0@NNN?N& P  ;5(2L  V  :#   2 ; Ho?"0@NNN?NS   < HA o?"0@NNN?N V  ;2(2L  C#  $ 2 D Ho?"0@NNN?N E H F o?"0@NNN?N ;2(2L J  F#   2 G Ho?"0@NNN?NZ H HJ o?"0@NNN?NJ  ;1(2H  0޽h ? zw33fYyy___PPT10.X4 >+iD' = @B D]' = @BA?%,( < +O%,( < +D' =%(D<' =%(Ds' =4@BB2BB%()?D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*LD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*LD' =-g6B fade*<3<*LDs' =4@BB2BB%()?D' =1:Bvisible*o3>+B#style.visibility<*M%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*MD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*MD' =-g6B fade*<3<*MDs' =4@BB2BB%()?D' =1:Bvisible*o3>+B#style.visibility<*X%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*XD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*XD' =-g6B fade*<3<*XDs' =4@BB2BB%()?D' =1:Bvisible*o3>+B#style.visibility<*c%(D' =+4 8?bCB#ppt_w+.3BCB#ppt_wB*Y3>B ppt_w<*cD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*cD' =-g6B fade*<3<*c+}  0 `$(  r  S X     r  S hY H    H  0޽h ? zw33fYyy___PPT10i.N6+D=' I = @B +  0 0 4(  r  S g }      s *A ? ?K8 $D  0  6j  } ,$  0 J(1) Start with a JL(n) coloring in which colors have frequency 1.^K""""5"&6`   6p   ,$  0 (2) If edge uv bears a color with frequency >1, then color all uv edge bundle in Kn() with that color.h ""0""""**"&> 1H  0޽h ? zw33fYyy  ___PPT10 .Y4pzk+-K/D+ ' I = @B D' = @BA?%,( < +O%,( < +D'' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* +p+0+0 ++0+ 0 +  0 JBP(  r  S < '+"    Z  6  <8< ,$  0 JL(n, ) coloring will use (n-1) - + = n-1+(-1) colors. Y&&&&& &&  &U  s *A ? ?  8 $@  0  s *A ? ? = 8 $@  0  s *A ? ? L 8 $D  0H  0޽h ? zw33fYyy3 + ___PPT10 .M6P^+4D ' I = @B Db ' = @BA?%,( < +O%,( < +D ' =%(DA ' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*+8+0+0 +  0 GG@\ܔ`WG(  r  S  M   B ]@ NDo?"0@NNN?NB ^ NDo?"0@NNN?NL  _# C  2 ` Ho?"0@NNN?N a H o?"0@NNN?N ;1(2L J  b# C, 2 c Ho?"0@NNN?NZ d H o?"0@NNN?NJ  ;1(2B h@ NDo?"0@NNN?N G`B i NDo?"0@NNN?Nt`L  j# qG2 k Ho?"0@NNN?N l HH o?"0@NNN?N ;1(2L J  m# 2 n Ho?"0@NNN?NZ o Hh o?"0@NNN?NJ  ;1(2B q@ HDo?"0@NNN?N[>B r HDo?"0@NNN?N]?B s@ HDo?"0@NNN?NZB t HDo?"0@NNN?N L =  y# 2 z Ho?"0@NNN?N M  { H o?"0@NNN?N=  ;5(2L wM %  |#  O2 } Ho?"0@NNN?NJ%  ~ H̽ o?"0@NNN?NwM   ;2(2L   %  # Y/ !2  Ho?"0@NNN?N  %   H o?"0@NNN?N    ;3(22  No?"0@NNN?N,  Nl o?"0@NNN?N ;2(2L & c  #  2  Ho?"0@NNN?N& c   H o?"0@NNN?N& P  ;1(2B @ HDo?"0@NNN?N5 U ,$D  0B  HDo?"0@NNN?NX)X,$D  0B @ HDo?"0@NNN?N X,$@  0B @ HDo?"0@NNN?N )X,$D  0B @ HDo?"0@NNN?N5 )X,$@   0B  HDo?"0@NNN?N U ,$@   0B  HDo?"0@NNN?N5 X,$@  0B  HDo?"0@NNN?NU )X,$D   0B  HDo?"0@NNN?N5  ,$@  0B  HDo?"0@NNN?NU X,$@   0l Mw  ,$D  02  6( MG > 2  6 mq > 2  6  p > 2  6T fp > 2  6 w@ > B @ NDo?"0@NNN?N ,$@(  0B  NDo?"0@NNN?N S9 ,$@+  0B @ NDo?"0@NNN?N29 ,$D&  0B @ NDo?"0@NNN?N2S ,$@#  0B @ NDo?"0@NNN?NS ,$@  0B  NDo?"0@NNN?N2 ,$@  0B  NDo?"0@NNN?N9 ,$@  0B  NDo?"0@NNN?N S ,$@  0B  NDo?"0@NNN?N2,$@  0B  NDo?"0@NNN?N 9 ,$@  0B @ HDo?"0@NNN?N ,$@'  0B ”@ HDo?"0@NNN?N ,$D)  0B Ô HDo?"0@NNN?N_,$@  0B Ĕ HDo?"0@NNN?N,$@  0B Ŕ HDo?"0@NNN?N  ,$@  0B Ɣ HDo?"0@NNN?Ng ,$@  0B ǔ HDo?"0@NNN?N: ,$@  0B Ȕ@ HDo?"0@NNN?N ,$@  0B ɔ@ HDo?"0@NNN?N& ,$@  0B ʔ HDo?"0@NNN?NZ ,$@  0B ˔ HDo?"0@NNN?N  ,$@  0B ̔ HDo?"0@NNN?N : ,$@  0B ͔ HDo?"0@NNN?N : ,$@  0B Δ@ HDo?"0@NNN?N8 ,$@$  0B ϔ@ HDo?"0@NNN?NT ,$@"  0B Д@ HDo?"0@NNN?N2 ,$@!  0B Ӕ@ HDo?"0@NNN?N2a ,$@%  0B Ԕ HDo?"0@NNN?N T: ,$@*  0B Ք HDo?"0@NNN?N T: ,$D,  0B ֔ HDo?"0@NNN?N & ,$D   0B ה HDo?"0@NNN?N ,$@  0  Mw # m  ,$D   02  6 MG > 2  6X mq > 2  6d p > 2  6 fp > 2  6l w@ > l  V   ܔV  ,$D-  0  ڔ 0(   V   *n-1+(-1) =8r&& &n ۔ s *A ? ?{ \  H  0޽h ? zw33fYyyjj___PPT10j._40$t+_̡Dbj' I = @B Dj' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D ' =%(D ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D*' =%(D*' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ô%(D' =-m6Bbox(in)*<3<*ÔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ĕ%(D' =-m6Bbox(in)*<3<*ĔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ŕ%(D' =-m6Bbox(in)*<3<*ŔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ɣ%(D' =-m6Bbox(in)*<3<*ƔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ǔ%(D' =-m6Bbox(in)*<3<*ǔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ȕ%(D' =-m6Bbox(in)*<3<*ȔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ɔ%(D' =-m6Bbox(in)*<3<*ɔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ʔ%(D' =-m6Bbox(in)*<3<*ʔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*˔%(D' =-m6Bbox(in)*<3<*˔D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*̔%(D' =-m6Bbox(in)*<3<*̔D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*͔%(D' =-m6Bbox(in)*<3<*͔D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ה%(D' =-m6Bbox(in)*<3<*הD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*֔%(D' =-m6Bbox(in)*<3<*֔D ' =%(D ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Д%(D' =-m6Bbox(in)*<3<*ДD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ϔ%(D' =-m6Bbox(in)*<3<*ϔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Δ%(D' =-m6Bbox(in)*<3<*ΔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ӕ%(D' =-m6Bbox(in)*<3<*ӔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*DS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*”%(D' =-m6Bbox(in)*<3<*”DS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ԕ%(D' =-m6Bbox(in)*<3<*ԔD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Ք%(D' =-m6Bbox(in)*<3<*ՔD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ܔ%(D' =-m6Bbox(in)*<3<*ܔ+q  0 p(  r  S PE Ffj     f  s *A ? ?.  H  0޽h ? zw33fYyy___PPT10i.O6p+D=' I = @B +  0 (  r  S @M    H  0޽h ? zw33fYyy___PPT10i.{7+D=' I = @B +  0 (  r  S lY     f  s *A ? ? f  s *A ? ?dz  H  0޽h ? zw33fYyy___PPT10i.|7+D=' I = @B +   0 91(  r  S Lg YP   2  Ho?"0@NNN?Nw   Nh o?"0@NNN?N  75 K = Kn()-v (2&.&&..  B  @ HDjJ?"0@NNN?N B  HDjJ?"0@NNN?N B @ HDjJ?"0@NNN?N B @ HDjJ?"0@NNN?N@ B  HDjJ?"0@NNN?N B  HDjJ?"0@NNN?N@  B @ HDo?"0@NNN?N ,$@  0B  HDo?"0@NNN?N ,$@  08 ct  2  6u cj >   Ny o?"0@NNN?Nt ;v(22   <}    > 2   6\    >   Bl o?"0@NNN?NA ,$  0 A new color (2   B o?"0@NNN?Nm  ,$  0 A new color (2 H  0޽h ? zw33fYyyx p ___PPT10P .7qF+l^D ' I = @B Do ' = @BA?%,( < +O%,( < +D ' =%(DN ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*+p+0+0 ++0+0 +  0 *(    s x 0e0e #" 0ez    H  0޽h ? zw33fYyy___PPT10i.7@+D=' I = @B +  0  t(  r  S     f  s *A ? ?  f  s *A ? ? 0  H o?"0@NNN?N"e <1, if n is odd , if n is even(0Z2022"r  NH]#o?"0@NNN?N}8H  0޽h ? zw33fYyy___PPT10i.7 L0+D=' I = @B +4  0 *(  r  S , Z"   2  Ho?"0@NNN?Nw   N o?"0@NNN?N   K = Kn()-v (2&.&&..  B @ HDjJ?"0@NNN?N B  HDo?"0@NNN?N ,$@   0B @ HDjJ?"0@NNN?N B  @ HDjJ?"0@NNN?N@ B   HDo?"0@NNN?N ,$@   0B   HDo?"0@NNN?N@  ,$D   0B  @ HDo?"0@NNN?N ,$@  0  B o?"0@NNN?NA ,$  0 A new color (2   T o?"0@NNN?N 2  ;u(2%  N o?"0@NNN?N T2 ,$  0 ;w(2B  HDf3o?"0@NNN?N  ,$D  0B  HDo?"0@NNN?N  ,$@  0B  HDo?"0@NNN?N  ,$@  02  6   ,$@  0 > B @ HDo?"0@NNN?N ,$@   0B @ HDo?"0@NNN?N? ,$D   0F ct   2  6 cj >   N o?"0@NNN?Nt ;v(22  6    > H  0޽h ? zw33fYyy___PPT10.7+xMD' I = @B D' = @BA?%,( < +O%,( < +D'' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D7 ' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*DS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*++0+0 ++0+0 ++0+0 +3  0 %(  r  S 8 _@   2  Ho?"0@NNN?Nw   N" o?"0@NNN?N   K = Kn()-v (2&.&&..  B @ HDjJ?"0@NNN?N B  HDo?"0@NNN?N ,$@  0B @ HDjJ?"0@NNN?N B  @ HDjJ?"0@NNN?N@ B   HDo?"0@NNN?N ,$@  0B   HDo?"0@NNN?N@  ,$@  0B  @ HDo?"0@NNN?N ,$@  0   B" o?"0@NNN?Nm ,$  0 A old color (2 B  HDo?"0@NNN?N  ,$D   0B  HDo?"0@NNN?N  ,$@   0B  HDo?"0@NNN?N  ,$@   02  6<"   ,$D  0 >   Bd" o?"0@NNN?N@ ,$  0 A new color (2 B @ HDo?"0@NNN?N ,$@   0B @ HDo?"0@NNN?N? ,$D   0F ct   2  6<" cj >   NX" o?"0@NNN?Nt ;v(22  6!"    > H  0޽h ? zw33fYyy___PPT10.706+Y7D' I = @B D' = @BA?%,( < +O%,( < +D'' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D7 ' =%(DA ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*DS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*++0+ 0 ++0+0 ++0+0 +PM  0 (: :(V9(  r  S =" @ "  2 $ Ho?"0@NNN?N L  % N,?" o?"0@NNN?Ni Cw  K = Kn()-v (2&.&&..  B &@ HDjJ?"0@NNN?N-{ B ,@ HDo?"0@NNN?N X  - B8J" o?"0@NNN?N  A new color (2  . N0O" o?"0@NNN?NH r  ;u(2 / NR" o?"0@NNN?N 0  ;w(2h8   @ Z " B 0 HDf3o?"0@NNN?N B 1 HDo?"0@NNN?N B 2 HDo?"0@NNN?N B 4@ HDo?"0@NNN?N-z B 5@ HDo?"0@NNN?N` vF   A   B B HDo?"0@NNN?N B C HDo?"0@NNN?N B D HDo?"0@NNN?N B F HDo?"0@NNN?N{ B G HDo?"0@NNN?N B H HDo?"0@NNN?NN B I HDo?"0@NNN?N{ C B J HDo?"0@NNN?NZ " B K HDo?"0@NNN?NM %  L  0e0e    BCDEFA o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN E5%  N E5%  N F   5%    !"?N@ABC DEFFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab -  M  0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab -  N  0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab4 - B O HDD[?"0@NNN?Nn  ,$D  0B P HDD[?"0@NNN?Nz ,$D  0B Q@ HDD[?"0@NNN?N> ,$D  0B R HDD[?"0@NNN?N ,$D  0 S  0e0e    BCDEF D[ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@`     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab - ,$D  0B T BDD[?"0@NNN?N 3  ,$D  0F ct 6 M ^ 2 7 64a" cj >  8 NPd" o?"0@NNN?Nt ;v(22 9 6h"  [,  > 2 ? 6k"  D+  > 2 3 6o"  +  > H  0޽h ? zw33fYyy___PPT10.7 y߿+KsD|' I = @B D7' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*O%(D' =-m6Bbox(in)*<3<*OD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =-m6Bbox(in)*<3<*PD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*Q%(D' =-m6Bbox(in)*<3<*QD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*R%(D' =-m6Bbox(in)*<3<*RD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*S%(D' =-m6Bbox(in)*<3<*SD' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*T%(D' =-m6Bbox(in)*<3<*T+_  0 @@/3D@(  2 . Ho?"0@NNN?N `,$D  0  S "  <$ 0 "  .  0x"   ,$  0 JL(n-1, ) coloring (using n-2 colors, and edge bundles are of frequency 1.) `___&___&___&______&___&___A___&___  s *A ? ?8 $D  02  Ho?"0@NNN?N a  N" o?"0@NNN?NS} K = Kn()-v (2&.&&..  B  @ HDjJ?"0@NNN?N-{ B  @ HDo?"0@NNN?N X    B" o?"0@NNN?N  A new color (2    Nȟ" o?"0@NNN?N] ;u(2   NP" o?"0@NNN?N0  ;w(2vF    Z 7B  HDf3o?"0@NNN?N B  HDo?"0@NNN?N B  HDo?"0@NNN?N B @ HDo?"0@NNN?N-z B @ HDo?"0@NNN?N` vF     (B  HDo?"0@NNN?N B  HDo?"0@NNN?N B  HDo?"0@NNN?N B  HDo?"0@NNN?N{ B  HDo?"0@NNN?N B  HDo?"0@NNN?NN B  HDo?"0@NNN?N{ CB  HDo?"0@NNN?NZ "B  HDo?"0@NNN?NM :   0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab-   0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab.-    0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abI-+B ! HD)?"0@NNN?Nn B " HD)?"0@NNN?Nz B #@ HD)?"0@NNN?N> B $ HD)?"0@NNN?N   %  0e0e    BCDEF ) 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab1-B & BDt?"0@NNN?N3 F ct ' M ^ )2 ( 6̰" cj >  ) N" o?"0@NNN?Nt ;v(22 * 6L" [A > 2 + 6 "  D@ > 2 , 6"  @ > B /@ ND___)?"0@NNN?N2,$D  0 0 s *A ? ?  8 $@ 0 2 s *A ? ?  8 $D  0 3 <L" )?"0@NNN?N,,$  0 ( JL(n, ) coloring  2&&& &H  0޽h ? zw33fYyy___PPT10x.9@l+D' " = @B D_' = @BA?%,( < +O%,( < +DI ' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-m6Bbox(in)*<3<*.D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*/%(D' =-m6Bbox(in)*<3<*/D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D ' =%(D ' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-6B'blinds(horizontal)*<3<*1DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*1m%(D' =-6B'blinds(horizontal)*<3<*1mDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =-6B'blinds(horizontal)*<3<*0DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-6B'blinds(horizontal)*<3<*2D?' =%(D' =%(D' =A@BB)BB0B@ A%(D' =1:Bvisible*o3>+B#style.visibility<*3%(DE' =+ 4 8?CB#ppt_xBCB#ppt_x+.1BCB#ppt_xB*Y3>B ppt_x<*3D' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*3DM' =+4 8?CB#ppt_h/10BCB#ppt_h+.01BCB#ppt_hB*Y3>B ppt_h<*3DM' =+4 8?CB#ppt_w/10BCB#ppt_w+.01BCB#ppt_wB*Y3>B ppt_w<*3D' =+B#0,0; .5, 1; 1, 1-g6B fade*<3<*3++0+0 ++0+0 ++0+30 +`  0 80%(    S L" ,<$D 0 "  2  Ho?"0@NNN?N 9 ,$@  0  Np" o?"0@NNN?Ni R ,$  0 K = Kn()-v (2&.&&..  B @ HDjJ?"0@NNN?N/ } F ,$@  0B  HDjJ?"0@NNN?N} SF ,$@  0B  @ HDjJ?"0@NNN?N\ F ,$@  0B   HDjJ?"0@NNN?NO %F ,$@  0B   HDjJ?"0@NNN?N F ,$@  0B  @ HDo?"0@NNN?N O F ,$@   0   Bt" o?"0@NNN?N ? ,$   0 A new color (2 %  N," o?"0@NNN?N / ,$  0 ;w(2B  HDjJ?"0@NNN?N$ \ %$ ,$@  0B  HDjJ?"0@NNN?NF \ %F ,$@  0B  HDjJ?"0@NNN?Ng \ %g ,$D  0)  H" )?"0@NNN?NB +l ,$  0 E No new colors(2___)  H$ )?"0@NNN?NE o ,$  0 E No new colors(2___B  HDo?"0@NNN?Na 7$ ,$@  0B  HDo?"0@NNN?N d$ ,$@  0B  HDo?"0@NNN?NO %E ,$@  0   Hh$ )?"0@NNN?N<v,$  0 &uv and uw bundles have the same color.L' vK$B # HDo?"0@NNN?NE ] &E ,$@  0B $ HDo?"0@NNN?Nh ] &h ,$@  0B % HDo?"0@NNN?N" ] &" ,$D  0)l  6 q`  !  ,$D   0  N$ o?"0@NNN?N 6 q`  ;u(22  6$  '   > 7z ct  2O ` \,$@  02  6$ cj >   N$ o?"0@NNN?Nt ;v(22  64$   ] t ,$@   0 > H  0޽h ? zw33fYyyCC___PPT10B.9*+TOD2A' I = @B D@' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6^%(D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*^p%(D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-m6Bbox(in)*<3<*!D ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D4' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D ' =%(D ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-m6Bbox(in)*<3<*#D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-m6Bbox(in)*<3<*$D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-m6Bbox(in)*<3<*%D' =%(D' =%(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-m6Bbox(in)*<3<* ++0+0 ++0+ 0 ++0+0 ++0+0 ++0+ ++0+ 0 ++0+0 +M  0 d\@P(  r  S (S$ ZY $  f  s *A ? ? >  f  s *A ? ?)  o  f P s *A  ? ?^    H  0޽h ? zw33fYyy___PPT10i.F?,++D=' I = @B +,  0 4T(  2  Ho?"0@NNN?N  ,$D 0.  0"e  ,,$  0 JL(n-1, ) coloring (using n-2 colors, and edge bundles are of frequency 1.) `___&___&___&______&___&___A___&___  s *A  ? ?) )o 8  $D  02  Ho?"0@NNN?N    Np$ o?"0@NNN?N   K = Kn()-v (2&.&&..  B  @ HDjJ?"0@NNN?N#-{ T B  @ HDo?"0@NNN?N# X T    N{$ o?"0@NNN?N  ;w(2B @ HDo?"0@NNN?N#-z T B @ HD3o?"0@NNN?N` A B  HDo?"0@NNN?N#{ T B  HDo?"0@NNN?N# T B  HDo?"0@NNN?N#Z T B  HDo?"0@NNN?N{ CT B  HDo?"0@NNN?NZ "` B  HDo?"0@NNN?NM { 2 * 6$ ' [  > B -@ ND___)?"0@NNN?N s ,$@  0n / N$ )?"0@NNN?N2@ Kn() Kn monochromatic edge bundle that color -polychromatic edge bundle a new color\ vK"**  "   $RB 1 HD)?"0@NNN?Nx [ ,$@  0B 3 HD)?"0@NNN?N"{ S ,$@  0B 4@ BD|?"0@NNN?N C > ,$D  02 , 6,$ &  > 2 + 6$ & D  > F ct ' @M ^ j2 ( 6$ cj >  ) NX$ o?"0@NNN?Nt ;v(2H  0޽h ? zw33fYyy___PPT10.M?3e+.TÍDp' $ = @B D+' = @BA?%,( < +O%,( < +DS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-m6Bbox(in)*<3<*1D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-m6Bbox(in)*<3<*3D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-m6Bbox(in)*<3<*4D ' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-m6Bbox(in)*<3<*-D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*+8+0+0 +]%  0 p!M(  r  S $  $  B @ HDo?"0@NNN?Nk B  HDjJ?"0@NNN?N B @ HD3o?"0@NNN?N B  HDjJ?"0@NNN?N a B  HDjJ?"0@NNN?N B  @ HDo?"0@NNN?N>    N$ o?"0@NNN?N-k?  ;w(2B   HDjJ?"0@NNN?Na B   HDjJ?"0@NNN?Na B   HDjJ?"0@NNN?Na B  HDo?"0@NNN?Nx s B  HDo?"0@NNN?Nx B  HDo?"0@NNN?N a B  HDo?"0@NNN?Nb B  HDo?"0@NNN?Nb B  HDo?"0@NNN?Nb F  6 q`   U ?   N$ o?"0@NNN?N 6 q`  ;u(22  6H$  '   > F ct   2  6H$ cj >   N$ o?"0@NNN?Nt ;v(22  6d$ > > B  HDt?"0@NNN?N ,$@  0B  HDg?"0@NNN?Nk ,$@  0B @ BD?"0@NNN?Nq ,$D  0P  NH$ )?"0@NNN?N  ,$ 0 fThen K3 is edge colored with (n-2)+1= 2 colors, and = 1 edges are of frequency 1. e vK !   s *A  ? ?{  8  $D 0H  0޽h ? zw33fYyyH@___PPT10 .K?H+öD ' $ = @B Dw ' = @BA?%,( < +O%,( < +DS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-m6Bbox(in)*<3<*DS' =%(D' =%(DT' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-6B'blinds(horizontal)*<3<* +8+0+0 +  0  `L(    S $ <$@ 0 $   H s *A ? ? 8 S 8 $@ 0 K s *A ? ? @ ^ 8 $@ 0 L s *A ? ? + I8 $D 0H  0޽h ? zw33fYyy3+___PPT10 .J?и+$$^D' $ = @B D' = @BA?%,( < +O%,( < +D' =%(Dy' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' =-6B'blinds(horizontal)*<3<*mDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D' =-6B'blinds(horizontal)*<3<*HDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D' =-6B'blinds(horizontal)*<3<*KDG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*L%(D' =-6B'blinds(horizontal)*<3<*L+  0 t`l`EF`(  x  c $0 ' V6  '  2  Ho?"0@NNN?N u !  N$ o?"0@NNN?N A  kG2 2&  B @ HDjJ?"0@NNN?N6 Xg B @ HDo?"0@NNN?N6 5g   N' o?"0@NNN?N   ;w(2B @ HDo?"0@NNN?N6 Wg B  @ HD3o?"0@NNN?N# =T B   HDo?"0@NNN?N6 Xg B   HDo?"0@NNN?N6 zg B   HDo?"0@NNN?N6 7tg B   HDo?"0@NNN?N X g ,$@ 0B  HDo?"0@NNN?N 7s ,$@ 0B  HDo?"0@NNN?N0 * ,$@ 0F ct  S *;} 2  6p' cj >   N' o?"0@NNN?Nt ;v(22  6' 9 h  >   N ' o?"0@NNN?N) gxS  ;u(2vF    N "h B  HDo?"0@NNN?N B  HDo?"0@NNN?N B  HDo?"0@NNN?N z -F ^   h ,$@ 0   0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab-F (    0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab-a C    0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab-| ^ %  NL&' o?"0@NNN?N  ,$ 0 ;x(2   B)'  P  '  2 ! Ho?"0@NNN?N Z,$@  0U " Nl+' o?"0@NNN?Nw,$  0 kG2 2&  B #@ HDjJ?"0@NNN?N6 g ,$@  0B $@ HDo?"0@NNN?N6 g ,$@  0% % N|0' o?"0@NNN?N m ~ ,$  0 ;w(2B &@ HDo?"0@NNN?N6 g ,$@ 0B '@ HD3o?"0@NNN?N# +T ,$@ 0B ( HDo?"0@NNN?N6 6g ,$@ 0B ) HDo?"0@NNN?N6 Xg ,$@ 0B * HDo?"0@NNN?N6 g ,$@ 0B + HDo?"0@NNN?N g ,$@ 0B , HDo?"0@NNN?N s ,$@ 0B - HDo?"0@NNN?N0  ,$@ 0% 4 Np8' o?"0@NNN?N< f ,$ 0 ;u(2z   5 N  ,$@ 0B 6 HDo?"0@NNN?N B 7 HDo?"0@NNN?N B 8 HDo?"0@NNN?N z -F ^  9 h ,$@ 0 :  0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab-F (  ;  0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab-a C  <  0e0e    BCDEF o 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||q7^q@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab-| ^ % = N?' o?"0@NNN?N  ,$ 0 ;x(2z   > N  ,$@ 0B ? HDjJ?"0@NNN?N B @ HDjJ?"0@NNN?N B A HDjJ?"0@NNN?N z   C I 6| ,$@ 0B D HDo?"0@NNN?N B E HDo?"0@NNN?N B F HDo?"0@NNN?N 2  6G' 9 ! ,$@ 0 > 2  6J' : 8  > 2 3 6N' 9 d ,$@ 0 > 2 2 6R' 9 g ,$@ 0 > 2 1 6U' : ~ ,$@ 0 > 7z ct . S } ,$D 02 / 6Z' cj >  0 N|T' o?"0@NNN?Nt ;v(2H  0޽h ? zw33fYyyFF___PPT10F.W?@I+hDD' $ = @B DID' = @BA?%,( < +O%,( < +D ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bcircle(in)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bcircle(in)*<3<*D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =-s6Bcircle(in)*<3<*>DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bcircle(in)*<3<*Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<* %(D' =-s6Bcircle(in)*<3<* Dg' =4@BBBB%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bcircle(in)*<3<*Dt' =A@BBBB0B%()))D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-s6Bcircle(in)*<3<*Dm2' =%(D2' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D' =-s6Bcircle(in)*<3<*!DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D' =-s6Bcircle(in)*<3<*"D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-s6Bcircle(in)*<3<*#D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-s6Bcircle(in)*<3<*$DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-s6Bcircle(in)*<3<*%D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*&%(D' =-s6Bcircle(in)*<3<*&D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-s6Bcircle(in)*<3<*'D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-s6Bcircle(in)*<3<*(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-s6Bcircle(in)*<3<*)D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-s6Bcircle(in)*<3<**D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-s6Bcircle(in)*<3<*+D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-s6Bcircle(in)*<3<*,D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*-%(D' =-s6Bcircle(in)*<3<*-DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =-s6Bcircle(in)*<3<*4D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*5%(D' =-s6Bcircle(in)*<3<*5D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =-s6Bcircle(in)*<3<*9DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D' =-s6Bcircle(in)*<3<*=D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =-s6Bcircle(in)*<3<*CDD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-s6Bcircle(in)*<3<*3DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*2%(D' =-s6Bcircle(in)*<3<*2DD' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-s6Bcircle(in)*<3<*1D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D' =-s6Bcircle(in)*<3<*.++0+0 ++0+"0 ++0+%0 ++0+40 ++0+=0 ++0+0 ++0+30 ++0+20 ++0+10 +~+  0 ,j(  2 % Ho?"0@NNN?N ,$D 0r  S P' N '  2  Ho?"0@NNN?N !  N' o?"0@NNN?N6 `  kG2 2&  B @ HDjJ?"0@NNN?NP,z B @ HDo?"0@NNN?NP W   NȢ' o?"0@NNN?N  ;w(2B  @ HDo?"0@NNN?NP,y B  @ HD3o?"0@NNN?N=_ n B   HDo?"0@NNN?NPz B   HDo?"0@NNN?NP B   HDo?"0@NNN?NPY   NT' o?"0@NNN?NC m  ;u(2vF    h D B  HDo?"0@NNN?N B  HDo?"0@NNN?N B  HDo?"0@NNN?N . ' 0L'   ,$  0 JL(n-2, ) coloring (using n-3 colors, and edge bundles are of frequency 1.) `___&___&___&______&___&___A___&___B (@ ND___)?"0@NNN?N j ,$@  0 ) s *A =? ? 1 8 =$D  0B * HD)?"0@NNN?NPz ,$D  0B + HD)?"0@NNN?N - ,$D  0B ,@ BDJ?"0@NNN?N\6v ,$D  02  6' S  > F ct  mL ] 2  6' cj >   N' o?"0@NNN?Nt ;v(22 $ 6|' T Z  > H  0޽h ? zw33fYyyLD___PPT10$.Y?| +W=pD' ' = @B D{' = @BA?%,( < +O%,( < +D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<**%(D' =-m6Bbox(in)*<3<**D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*+%(D' =-m6Bbox(in)*<3<*+D' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*,%(D' =-m6Bbox(in)*<3<*,D ' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(D' =-s6Bcircle(in)*<3<*%D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =-m6Bbox(in)*<3<*(D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-m6Bbox(in)*<3<*'D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =-m6Bbox(in)*<3<*)+8+0+'0 +  0 Q(    6'  c  15. Forbidding rainbow directed triples in Dn 62x..H  0޽h ? zw33fYyy___PPT10i.hA@\+D=' ' = @B +   0   L (  r  S ' P  '  B @ ZDԔ?"0@NNN?N+l B  ZDԔ?"0@NNN?Nu w B  ZDԔ?"0@NNN?N4u 2  6' H   > 2  6e P > 2  6' H   > B @ ZDԔ?"0@NNN?N Y B   fDԔ?"0@NNN?Nb  c B  ZDԔ?"0@NNN?N!b 2  6' 5   > 2  6' = > 2  6' 5   >   H- )?"0@NNN?N M B  G Cyclic triple(2  Hd- )?"0@NNN?N } 9B  KTransitive triple(2H  0޽h ? zw33fYyy___PPT10i.hAL+D=' ' = @B +  0  \(  \r \ S  - B  -  r \ S  -   -  f \ s *A ? ? @   H \ 0޽h ? zw33fYyy___PPT10i.Bp'1+D=' ' = @B +  0 (  r  S - }  -  r  S - 2 -  f  s *A @? ?G @H  0޽h ? zw33fYyy___PPT10i.mA+D=' ' = @B +  0 (  r  S X-- V  -  f  s *A ? ?& X l f  s *A  ? ?lJ  H  0޽h ? zw33fYyy___PPT10i.mA0+D=' ' = @B +1  0  $h(  $ $ S ;- <$@ 0 -   $ s *A  ? ?U8  $@ 0 $ s *A ? ?WK 8 $D 0  $ s *A ? ? ^ 8 $@  0  $ s *A ? ? K ` 8 $D  0B  $ HDo?"0@NNN?N,$@  0>  $ H@- jJ?"0@NNN?NT0 ,$  0 Z# of colors in D.(2H $ 0޽h ? zw33fYyy___PPT10.BP{t+=zDu' ' = @B D0' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$3%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$3`%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D'' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-m6Bbox(in)*<3<* $D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =-m6Bbox(in)*<3<* $DX' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<* $%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<* $%(+8+0+ $0 +'  0   'h (  h h S HZ- <$@  0 -   h s *A ? ?Y0 8 $@  0f h s *A ? ?"Dh .2 h T\- o?"0@NNN?NF ,$@  0 >(2 X h N_- o?"0@NNN?N ,1 ,$  0 nD2 2&  B  h@ NDjJ?"0@NNN?N2 % ,$@  0%  h N e- o?"0@NNN?N " ,$  0 ;u(27z ct h %6,$@  02 h 6i- cj >  h Nl- o?"0@NNN?Nt ;v(22 h 6q-  3 ,$@  0 > B  h@ NDjJ?"0@NNN?N ^ ,$@  0f &h s *A ? ?  'h s *A ? ?I8 $D  0H h 0޽h ? zw33fYyy___PPT10q./CP.a+ gDe' ' = @B D ' = @BA?%,( < +O%,( < +Dk' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =-m6Bbox(in)*<3<* hD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =-m6Bbox(in)*<3<* hD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<* h%(D' =-m6Bbox(in)*<3<* hD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-m6Bbox(in)*<3<*hD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-m6Bbox(in)*<3<*hD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-m6Bbox(in)*<3<*hD>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-m6Bbox(in)*<3<*hD' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*h,%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =4@BBBB%())))?D' =1:Bvisible*o3>+B#style.visibility<*'h%(++0+h0 ++0+h0 ++0+ h0 ++0+h0 +H  0 r#j#@ 0p#(  px p c $<4   2 p Ho?"0@NNN?NS $ p No?"0@NNN?N c  nD2 2&  B p@ NDjJ?"0@NNN?NqwjE  p N?o?"0@NNN?N   ;u(2B p NDjJ?"0@NNN?N[ 0 ,$@  0% p NTDo?"0@NNN?NS } ,$  0 ;w(22 p 6H6  ,$@  0 > F ct p Pj{ z2 p 6Mcj >  p NPo?"0@NNN?Nt ;v(22  p 6$U7 x  > B !p NDjJ?"0@NNN?N` 5 ,$@  0B "p@ NDjJ?"0@NNN?N@bU B #p NDjJ?"0@NNN?NK  K ,$@  0B $p NDjJ?"0@NNN?Nt  t ,$D  0% 'p NYo?"0@NNN?N ` q ,$  0 ;x(22 (p 6L6 3 ,$D   0 > B )p NDjJ?"0@NNN?NS   S ,$@  0B *p NDjJ?"0@NNN?N|   | ,$D  0 +p  0e0e    BCDEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||2ZdZ@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab R; N ,$D   0 ,p  0e0e    BCDEF jJ 8c8c     ?1 d0u0@Ty2 NP'p<'pA)BCD|E||2ZdZ@  "0e@     @ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `abN 5%  N 5%  N    5%    !"?N@ABC DEFGHIJK5%LMNOPQRSTUWYZ[ \]^_ `ab R; { ,$@   0B -p NDjJ?"0@NNN?Nx. ! ,$@   0B .p NDjJ?"0@NNN?NH3 ,$@   0H p 0޽h ? 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