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Then, the minimum number of colors needed to properly color G, c(G), is either D(G) or D(G) + 1. (Vizing s Theorem) G is of class one if c(G) = D(G) and class two otherwise. Kn is of class one if and only if n is even. Kn,n is of class one. (,j +///FE+> Rainbow Subgraphr/ vLet G be an edge-colored graph. Then a subgraph whose edges are of distinct colors is called a rainbow subgraph of G. It is also known as a heterochromatic subgraph or a multicolored subgraph. Note that the edge-coloring of the edge-colored graph may not be proper. In this talk, all edge-colorings are proper edge-colorings. Therefore, a rainbow star can be found easily.wP` LL P(8-?!Rainbow 1-factorTheorem (Woolbright and Fu, JCD 1998) In any (2m-1)-edge-colored K2m where m > 2, there exists a rainbow 1-factor. Conjecture (Transversal) In any n-edge-colored Kn,n, there exists a rainbow 1-factor for each odd integer n.   %  ;, ;C% Room Squares A Room square of side 2m-1 provides a (2m-1)-edge-coloring of K2m such that 2m-1 edge- disjoint multicolored 1-factors exist. 35 17 28 46 26 48 15 37 13 57 68 24 47 16 38 25 58 23 14 67 12 78 56 34 36 45 27 18 ~ZZZ? Z f /A#Orthogonal Latin SquaresdA Latin square of order n corresponds to an n-edge-coloring of Kn,n. A Latin square of order n with an orthogonal mate provides n edge-disjoint multicolored 1-factors of Kn,n. 1 2 3 1 2 3 2 3 1 3 1 2 3 1 2 2 3 1 nZZ@ h #   ,?g@"Multicolored Subgraph  Conjecture Given an (n-1)-edge-colored Kn for even n > n0, a multicolored Hamiltonian path exists.(By whom?) How about multicolored spanning trees?     )<'2 0DBrualdi-Hollingsworth s Conjecture"#$ $ If m>2, then in any proper edge coloring of K2m with 2m-1 colors, all edges can be partitioned into m multicolored spanning trees. P3iB$Multicolored Tree ParallelismX  K2m admits a multicolored tree parallelism (MTP) if there exists a proper (2m-1)-edge-coloring of K2m for which all edges can be partitioned into m isomorphic multicolored spanning trees.$ $ $$$$$ $#$ $/$ $$$D&K6 admits an MTP$ T1 T2 T3 Color 1: 35 46 12 Color 2: 24 15 36 Color 3: 25 34 16 Color 4: 26 13 45 Color 5: 14 23 560         $ $(((( ((Two Conjectures on MTP Constantine s Weak Conjecture For any natural number m > 2, there exists a (2m-1)-edge-coloring of K2m for which K2m can be decomposed into m multicolored isomorphic spanning trees. $$$*$ $ $ $$$($>Constantine s Strong Conjecture $ If m > 2, then in any proper edge coloring of K2m with 2m-1 colors, all edges can be partitioned into m isomorphic multicolored spanning trees. (Three, so far!) n48   $Theorem (Akbari, Alipour, Fu and Lo)8%   ,   For m is an odd positive integer, then K2m admits an MTP. Fact. Constantine s Weak Conjecture is true.Hl+)The Idea An m-total-coloring of a graph G is a mapping from V(G)E(G) into {1,2,3, & , m} such that incident vertices and edges receive distinct images.D$W$ An example: m = 5(P %An example: m = 5Z H'An example: m = 5Z I(An example: m = 5Z *An example: m = 5 -An example: m = 5 1An example: m = 5P 2An example: m = 5 3An example: m = 5 4An example: m = 5Z N+Edge-colored Kn, n is Odd*   It is well known that Kn is of class 2 when n is odd, i. e. the chromatic index of Kn is n. In order to find multicolored parallelism, each subgraph has to be of size n. The best candidate is therefore the Hamiltonian cycles of Kn..><7TQ.MHCPD K2m+1 admits a multicolored Hamiltonian cycle parallelism (MHCP) if there exists a proper (2m+1)-edge-coloring of K2m+1 for which all edges can be partitioned into m multicolored Hamiltonian cycles. xFFN FF FF FFFF FFFFFFFFN-FFFFFR/ The Existence@ Theorem (Constantine, SIAM DM) If n is an odd prime, then Kn admits an MHCP. Conjecture Kn admits an MHCP for each odd integer n. Yx-xFFFF%FFFFNFF0FN(F4:(T0MHCPDLemma (Fu and Lo, DM) Let v be a composite odd integer and n is the smallest prime which is a factor of v, say v = mn. If Km admits an MHCP, then Km(n) admits an MHCP."xF F F F F ,F F F F F F N F F N F ,sZ -Example: K9 Starts from K3(3). FFNN FFN[ Example: K9 jP FFNNN\ Example: K9 jP FFNNN] Example: K9 jP FFNNN^ Example: K9 jP FFNNN_ Example: K9 jP FFNNN` Example: K9 jP FFNNNaMHCP@CExample: K9 Use C1 + 3K3 to construct two Hamiltonian cycles :D F$F$N$F$FF$F$N$F$N$&F$N$b Example: K9 jP FFNNNc Example: K9 jP FFNNNd Example: K9 jP FFNNNe Example: K9 jP FFNNNjNot Many Are found in GeneralF(:Let  be an arbitrary (2m-1)-edge-coloring of K2m. Then there exist three isomorphic multicolored spanning trees in K2m for m > 2. Joint work with Y.H. Lo. xDD Dt5DDDDLDD+DDLDDDDDDDDFFJ) Observation If the edge-coloring is arbitrarily given, then the proof of finding MTP is going to be very difficult. If we drop  isomorphism for the above case, then can we find m multicolored spanning trees of K2m?<h` 1hProblem: How many multicolored spanning trees of an edge-colored K2m can we find if m is getting larger?8i(;( ($((Guess? Of course, the best result is m.4)(  Definition  (x(y) = c. x-1 (c) = y. x9 c: denotes the edge xy. x-star : Sx NT(x)(E     """"""*"*"""Z  Definition  ( Assume T is a multicolored spanning tree of K2m with two leaves x1, x2. Let the edges incident to x1 and x2 be e1 and e2 respectively, and (e1) = c1, (e2) = c2. Define T[x1,x2] = T  {e1,e2} + {x19 c2: , x29 c1: }. $           &&.& &  &  .  &  Definition  ( T is a set of mutually disjoint multicolored spanning trees in a (2m-1)-edge-colored K2m. A multicolored spanning tree T is said to be feasible for T if the union of trees in T and T are disjoint. cC  GCConstructing Process(Pick u U, and let (uxi ) = i . Choose v1 U \ {u} s.t. T1[u,v1] is feasible for T \T1. For 2 d" I d" n, choose vi U \ {u} s.t. Ti [u,vi] is feasible for T \Ti and (1) & , (2)& , (3)& & Redefine Ti = Ti [u,vi] and let Tn+1 = Su [u1,u2,& ,un].B"       C  c     C #           t H?"Constructing Process( Constructing Process( Constructing Process( Constructing Process( Note( Counting  (Let an = |V(K2m) - U|. U = a2 d" 7. an d" an-1 + 1 + 4(n-1). ! an d" 2n2-n+1.vPC C K C C K C C G G C K  c c k k c c k c c c c c c c c k c g o g c M*Theorem (with Y.H. Lo)*((oFor m large enough, we can find at least about (2m)1/2 multicolored spanning trees in (2m-1)-edge-colored K2m. np/$$$5$$$Don t Stop! 6Z &We Have to Stop! 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3333___PPT10i.v =_+D=' B= @B +   0   0<3 (  <x < c $,*`P  *  < c $d2*  * "p`Pp < 0l8* n /  OA 4-total-coloring of 5-cycle 2^ < 68c R2  < s * R2  < s * X R2  < s * p@R2  < s * X0R2  < s * 0  < 0lJH x/  31 2 < 0H?* o  31 2 < 0C*X ` ?  31 2 < 0XG*( 0  32 2 < 0K*   32 2 < 0A* 0p 32 2 < 0Q*0 X  33 2 < 0pU*( X  33 2 < 0Y* h 33 2 < 0]*  34 2H < 0޽h ? 3333___PPT10i.v@4.+D=' B= @B +a  0 @@@tׄp'@(  x  c $c*2P  *   c $@f*<$D 0 *  e 0g*F ,$D 0 i7Bipartite difference : 1 (Use a color for these edges.)8 28&z ]pc  g p]c ,$D 0t2 h 6"`]pt2 i 6"`pjt2 j 6"` p t2 k 6"`V p t2 l 6"`pYTB m c $D+] TB n c $DO ] TB o c $D s T TB p c $D  9 TB q c $D 9 t2 r 6"`g8  t2 s 6"`8 t t2 t 6"` 8  t2 u 6"`` 8 t2 v 6"` 8 c fz ]}` w }]`,$D 0 ]CY x# ]pY,$D0t2 y 6"`]Ct2 z 6"`Cjt2 { 6"` C t2 | 6"`V C t2 } 6"`CY  gc  ~# ^}Z,$D0n2  0"`g  n2  0"` t n2  0"`  n2  0"`` n2  0"`  c ZB  s *DԔj ZB  s *DԔ  ZB  s *DԔ `   3 B CDEFԔP  @  "`6   3 BCDEFԔ9qs@  "``   3 BQCDEFԔb1{^WNQ@  "` S 6   3 BCDEFԔb1zN" N@  "`jL ` ZB  s *DԔ    3 B#CDEFԔNS1#b@  "`S   3 BCDEFԔPu+e@  "`j  T ]CY # ]qYt2  6"`]Ct2  6"`Cjt2  6"` C t2  6"`V C t2  6"`CYt2  6"`^pt2  6"`pkt2  6"` p t2  6"`W p t2  6"`pZZB  s *DԔa ZB  s *DԔ ZB  s *DԔ W   C B CDEFԔP  @  "`}-p  C BCDEFԔ9qs@  "`W p  C BQCDEFԔb1{^WNQ@  "` -  C BCDEFԔb1zN" N@  "`aW ZB  s *DԔ  C B#CDEFԔNS1#b@  "`   C BCDEFԔPu+e@  "`aZ2  s *`z g9 h  9 gh,$D 0  9 5  # g9 c,$D0n2  0"`9  n2  0"` F n2  0"` n2  0"`2 n2  0"` 5 ZB  s *DԔ< ZB  s *DԔ  ZB  s *DԔ 2   C B CDEFԔP  @  "`g    C BCDEFԔ9qs@  "` 2   C BQCDEFԔb1{^WNQ@  "` j   C BCDEFԔb1zN" N@  "`< 2 j ZB  s *DԔ    C B#CDEFԔNS1#b@  "`g j   C BCDEFԔPu+e@  "`<  t2  6"`g9  t2  6"`9 t t2  6"` 9  t2  6"`` 9 t2  6"` 9 c ZB  s *DԔjf f ZB  s *DԔ f f ZB  s *DԔ f ` f   C B CDEFԔP  @  "` 6  C BCDEFԔ9qs@  "` ` v  C BQCDEFԔb1{^WNQ@  "` f 6   C BCDEFԔb1zN" N@  "`jf ` ZB  s *DԔf f  „ C B#CDEFԔNS1#b@  "`f  Ä C BCDEFԔPu+e@  "`jf }Z2 Ą s *@ h lz -9  ń -9 ,$D 0ZB Ƅ s *DԔ9 ZB DŽ s *DԔ==9 ZB Ȅ s *DԔ  9 ZB Ʉ s *DԔ  9 ZB ʄ s *DԔ--9 |l `Xp фh`X,$D 0`2 ˄ 0```2 ̄ 0Ph`2 ̈́ 0  p`2 ΄ 0`  p`2 τ 0Xhlz `Xp ҄ ( ph ,$D 0Z2 ӄ s *``Z2 Ԅ s *PhZ2 Մ s *  pZ2 ք s *`  pZ2 ׄ s *XhH  0޽h ? 3333  ___PPT10 ..v׀++3^DM ' *= @B D ' = @BA?%,( < +O%,( < +Dn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*w%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*wD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*wDn' =%(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<*ń%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*ńD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*ńDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ф%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*фD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*фDn' =%(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{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*eD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*eDn' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D' =+4 8?dCB0-#ppt_w/2BCB#ppt_xB*Y3>B ppt_x<*gD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*g+8+0+e0 +-*  0 D)<)PM ((  x  c $t*`P  * x  c $L* *  . 0*F  PBipartite difference : 1 and 2 2&8 ]}c }]cz2 0 <"`]qz2 1 <"`qjz2 2 <"` q z2 3 <"`V q z2 4 <"`qYz2 5 <"`^pz2 6 <"`pkz2 7 <"` p z2 8 <"`W p z2 9 <"`pZ`B : 0DԔa `B ; 0DԔ `B < 0DԔ W  = C B CDEFԔP  @  "`}-p > C BCDEFԔ9qs@  "`W p ? C BQCDEFԔb1{^WNQ@  "` - @ C BCDEFԔb1zN" N@  "`aW `B A 0DԔ B C B#CDEFԔNS1#b@  "`  C C BCDEFԔPu+e@  "`az2 D <"`g9  z2 E <"`9 t z2 F <"` 9  z2 G <"`` 9 z2 H <"` 9 c `B I 0DԔjf f `B J 0DԔ f f `B K 0DԔ f ` f  L C B CDEFԔP  @  "` 6 M C BCDEFԔ9qs@  "` ` v N C BQCDEFԔb1{^WNQ@  "` f 6  O C BCDEFԔb1zN" N@  "`jf ` `B P 0DԔf f  Q C B#CDEFԔNS1#b@  "`f  R C BCDEFԔPu+e@  "`jf }`B S 0DԔ9 `B T 0DԔ==9 `B U 0DԔ  9 `B V 0DԔ  9 `B W 0DԔ--9 T ]pc  X# ]pc t2 Y 6"`]pt2 Z 6"`pjt2 [ 6"` p t2 \ 6"`V p t2 ] 6"`pYTB ^ c $D+] TB _ c $DO ] TB ` c $D s T TB a c $D  9 TB b c $D 9 t2 c 6"`g8  t2 d 6"`8 t t2 e 6"` 8  t2 f 6"`` 8 t2 g 6"` 8 c . ]pc   ]pc ,$D 0t2  6"`]pt2  6"`pjt2  6"` p t2  6"`V p t2  6"`pYt2  6"`g8  t2  6"`8 t t2  6"` 8  t2  6"`` 8 t2  6"` 8 c TB  c $D ] TB  c $DCf Z TB  c $D T TB B c $Df 9 TB B c $DI9 H  0޽h ? 3333___PPT10i.w+D=' B= @B +0  0 //`]]h/(  ~  s **`P  * ~  s ** *   0*F  T"Bipartite difference : 1, 2, and 3# 2#h-8 ]}p ]}]pt2  6"`]qt2  6"`qjt2  6"` q t2  6"`V q t2   6"`qYt2   6"`^pt2   6"`pkt2   6"` p t2   6"`W p t2  6"`pZZB  s *DԔa ZB  s *DԔ ZB  s *DԔ W   C B CDEFԔP  @  "`}-p  C BCDEFԔ9qs@  "`W p  C BQCDEFԔb1{^WNQ@  "` -  C BCDEFԔb1zN" N@  "`aW ZB  s *DԔ  C B#CDEFԔNS1#b@  "`   C BCDEFԔPu+e@  "`at2  6"`g9  t2  6"`9 t t2  6"` 9  t2  6"`` 9 t2  6"` 9 c ZB  s *DԔjf f ZB  s *DԔ f f ZB   s *DԔ f ` f  ! C B CDEFԔP  @  "` 6 " C BCDEFԔ9qs@  "` ` v # C BQCDEFԔb1{^WNQ@  "` f 6  $ C BCDEFԔb1zN" N@  "`jf ` ZB % s *DԔf f  & C B#CDEFԔNS1#b@  "`f  ' C BCDEFԔPu+e@  "`jf }ZB ( s *DԔ9 ZB ) s *DԔ==9 ZB * s *DԔ  9 ZB + s *DԔ  9 ZB , s *DԔ--9 T ]pc  -# ]pc t2 . 6"`]pt2 / 6"`pjt2 0 6"` p t2 1 6"`V p t2 2 6"`pYTB 3 c $D+] TB 4 c $DO ] TB 5 c $D s T TB 6 c $D  9 TB 7 c $D 9 t2 8 6"`g8  t2 9 6"`8 t t2 : 6"` 8  t2 ; 6"`` 8 t2 < 6"` 8 c . ]pc  = ]pc ,$D 0t2 > 6"`]pt2 ? 6"`pjt2 @ 6"` p t2 A 6"`V p t2 B 6"`pYt2 C 6"`g8  t2 D 6"`8 t t2 E 6"` 8  t2 F 6"`` 8 t2 G 6"` 8 c TB H c $D ] TB I c $DCf Z TB J c $D T TB KB c $Df 9 TB LB c $DI9 4 ]pc  M# jhp ,$D 0t2 N 6"`]pt2 O 6"`pjt2 P 6"` p t2 Q 6"`V p t2 R 6"`pYt2 S 6"`g8  t2 T 6"`8 t t2 U 6"` 8  t2 V 6"`` 8 t2 W 6"` 8 c TB X c $D` K TB Y c $D= L TB Z c $D ? TB [ c $DIl 9 TB \ c $D 9 H  0޽h ? 3333___PPT10i.w+D=' B= @B +7  0 77pmm6(  ~  s **`P  * ~  s ** *   0*F  V$Bipartite difference : 1, 2, 3 and 4% 2%48 ]}p m}]pt2  6"`]qt2  6"`qjt2  6"` q t2  6"`V q t2   6"`qYt2   6"`^pt2   6"`pkt2   6"` p t2   6"`W p t2  6"`pZZB  s *DԔa ZB  s *DԔ ZB  s *DԔ W   C B CDEFԔP  @  "`}-p  C BCDEFԔ9qs@  "`W p  C BQCDEFԔb1{^WNQ@  "` -  C BCDEFԔb1zN" N@  "`aW ZB  s *DԔ  C B#CDEFԔNS1#b@  "`   C BCDEFԔPu+e@  "`at2  6"`g9  t2  6"`9 t t2  6"` 9  t2  6"`` 9 t2  6"` 9 c ZB  s *DԔjf f ZB  s *DԔ f f ZB   s *DԔ f ` f  ! C B CDEFԔP  @  "` 6 " C BCDEFԔ9qs@  "` ` v # C BQCDEFԔb1{^WNQ@  "` f 6  $ C BCDEFԔb1zN" N@  "`jf ` ZB % s *DԔf f  & C B#CDEFԔNS1#b@  "`f  ' C BCDEFԔPu+e@  "`jf }ZB ( s *DԔ9 ZB ) s *DԔ==9 ZB * s *DԔ  9 ZB + s *DԔ  9 ZB , s *DԔ--9 T ]pc  -# ]pc t2 . 6"`]pt2 / 6"`pjt2 0 6"` p t2 1 6"`V p t2 2 6"`pYTB 3 c $D+] TB 4 c $DO ] TB 5 c $D s T TB 6 c $D  9 TB 7 c $D 9 t2 8 6"`g8  t2 9 6"`8 t t2 : 6"` 8  t2 ; 6"`` 8 t2 < 6"` 8 c . ]pc  = ]pc ,$D 0t2 > 6"`]pt2 ? 6"`pjt2 @ 6"` p t2 A 6"`V p t2 B 6"`pYt2 C 6"`g8  t2 D 6"`8 t t2 E 6"` 8  t2 F 6"`` 8 t2 G 6"` 8 c TB H c $D ] TB I c $DCf Z TB J c $D T TB KB c $Df 9 TB LB c $DI9 4 ]pc  M# jhp ,$D 0t2 N 6"`]pt2 O 6"`pjt2 P 6"` p t2 Q 6"`V p t2 R 6"`pYt2 S 6"`g8  t2 T 6"`8 t t2 U 6"` 8  t2 V 6"`` 8 t2 W 6"` 8 c TB X c $D` K TB Y c $D= L TB Z c $D ? TB [ c $DIl 9 TB \ c $D 9 . ]pc  ] ]hc ,$D 0t2 ^ 6"`]pt2 _ 6"`pjt2 ` 6"` p t2 a 6"`V p t2 b 6"`pYt2 c 6"`g8  t2 d 6"`8 t t2 e 6"` 8  t2 f 6"`` 8 t2 g 6"` 8 c TB h c $D f TB iB c $D59 TB jB c $D= 9 TB kB c $D n 9 TB lB c $D 9 H  0޽h ? 3333___PPT10i.w+D=' B= @B +)'  0 @&8&IK%(  x  c $!.`P  .   B`". b . b!8 ]m G]mz2  <"`]qz2  <"`qjz2  <"` q z2   <"`V q z2   <"`qYz2   <"`^pz2   <"`pkz2   <"` p z2  <"`W p z2  <"`pZ  C BCDEFԔ9qs@  "`W p`B  0DԔz2  <"`g9  z2  <"`9 t z2  <"` 9  z2  <"`` 9 z2  <"` 9 c   C B CDEFԔP  @  "` 6  C B#CDEFԔNS1#b@  "`f `B  0DԔ9 z2  <"`]pz2  <"`pjz2  <"` p z2  <"`V p z2  <"`pYZB  s *D+] z2   <"`g8  z2 ! <"`8 t z2 " <"` 8  z2 # <"`` 8 z2 $ <"` 8 c z2 % <"`]pz2 & <"`pjz2 ' <"` p z2 ( <"`V p z2 ) <"`pYz2 * <"`g8  z2 + <"`8 t z2 , <"` 8  z2 - <"`` 8 z2 . <"` 8 c ZB /B s *DI9 z2 0 <"`]pz2 1 <"`pjz2 2 <"` p z2 3 <"`V p z2 4 <"`pYz2 5 <"`g8  z2 6 <"`8 t z2 7 <"` 8  z2 8 <"`` 8 z2 9 <"` 8 c ZB : s *D` K z2 ; <"`gpz2 < <"`ptz2 = <"` p z2 > <"`` p z2 ? <"` pcz2 @ <"`q8  z2 A <"`$8 ~ z2 B <"` 8  z2 C <"`j 8 z2 D <"`8 m ZB EB s *DG 9  F 0p7. Fs  8Tree 1 2 H 0;.   31 2 I 0?. h  32 2 J 0xC. @  33 2 K 0$G.p X x W  34 2H  0޽h ? 3333___PPT10i.LwM+D=' B= @B +$  0 *#"#EG"(  x  c $T.`P  . x  c $|U.b .  8 ]m  G]m z2  <"`]qz2  <"`qjz2  <"` q z2  <"`V q z2   <"`qYz2   <"`^pz2   <"`pkz2   <"` p z2   <"`W p z2  <"`pZ`B  0DԔa   C BCDEFԔPu+e@  "`az2  <"`g9  z2  <"`9 t z2  <"` 9  z2  <"`` 9 z2  <"` 9 c   C BCDEFԔb1zN" N@  "`jf ` `B  0DԔf f `B  0DԔ==9 z2  <"`]pz2  <"`pjz2  <"` p z2  <"`V p z2  <"`pYZB  s *DO ] z2  <"`g8  z2   <"`8 t z2 ! <"` 8  z2 " <"`` 8 z2 # <"` 8 c z2 $ <"`]pz2 % <"`pjz2 & <"` p z2 ' <"`V p z2 ( <"`pYz2 ) <"`g8  z2 * <"`8 t z2 + <"` 8  z2 , <"`` 8 z2 - <"` 8 c ZB . s *D ] z2 / <"`]pz2 0 <"`pjz2 1 <"` p z2 2 <"`V p z2 3 <"`pYz2 4 <"`g8  z2 5 <"`8 t z2 6 <"` 8  z2 7 <"`` 8 z2 8 <"` 8 c ZB 9 s *D= L z2 : <"`gpz2 ; <"`ptz2 < <"` p z2 = <"`` p z2 > <"` pcz2 ? <"`q8  z2 @ <"`$8 ~ z2 A <"` 8  z2 B <"`j 8 z2 C <"`8 m ZB DB s *D x 9  F 0i. Fs  8Tree 2 2H  0޽h ? 3333___PPT10i.Nw0E+D=' B= @B +$  0 *#"#EF"(  x  c $Pw.`P  . x  c $(x.: .  8 ]m  F]m z2  <"`]qz2  <"`qjz2  <"` q z2  <"`V q z2   <"`qYz2   <"`^pz2   <"`pkz2   <"` p z2   <"`W p z2  <"`pZ`B  0DԔ W   C B#CDEFԔNS1#b@  "` z2  <"`g9  z2  <"`9 t z2  <"` 9  z2  <"`` 9 z2  <"` 9 c `B  0DԔjf f   C BQCDEFԔb1{^WNQ@  "` f 6 `B  0DԔ  9 z2  <"`]pz2  <"`pjz2  <"` p z2  <"`V p z2  <"`pYZB  s *D s T z2  <"`g8  z2   <"`8 t z2 ! <"` 8  z2 " <"`` 8 z2 # <"` 8 c z2 $ <"`]pz2 % <"`pjz2 & <"` p z2 ' <"`V p z2 ( <"`pYz2 ) <"`g8  z2 * <"`8 t z2 + <"` 8  z2 , <"`` 8 z2 - <"` 8 c ZB . s *DCf Z z2 / <"`]pz2 0 <"`pjz2 1 <"` p z2 2 <"`V p z2 3 <"`pYz2 4 <"`g8  z2 5 <"`8 t z2 6 <"` 8  z2 7 <"`` 8 z2 8 <"` 8 c ZB 9 s *D ? z2 : <"`gpz2 ; <"`ptz2 < <"` p z2 = <"`` p z2 > <"` pcz2 ? <"`q8  z2 @ <"`$8 ~ z2 A <"` 8  z2 B <"`j 8 z2 C <"`8 m ZB DB s *D (9  E 0T. Fs  8Tree 3 2H  0޽h ? 3333___PPT10i.OwpO-+D=' B= @B +$  0 *#"#EF"(  x  c $.`P  . x  c $l. .  8 ]mv F]mvz2  <"`]qz2  <"`qjz2  <"` q z2  <"`V q z2   <"`qYz2   <"`^pz2   <"`pkz2   <"` p z2   <"`W p z2  <"`pZ`B  0DԔ   C BCDEFԔb1zN" N@  "`aW z2  <"`g9  z2  <"`9 t z2  <"` 9  z2  <"`` 9 z2  <"` 9 c `B  0DԔ f ` f   C BCDEFԔ9qs@  "` ` v`B  0DԔ  9 z2  <"`]pz2  <"`pjz2  <"` p z2  <"`V p z2  <"`pYZB  s *D  9 z2  <"`g8  z2   <"`8 t z2 ! <"` 8  z2 " <"`` 8 z2 # <"` 8 c z2 $ <"`]pz2 % <"`pjz2 & <"` p z2 ' <"`V p z2 ( <"`pYz2 ) <"`g8  z2 * <"`8 t z2 + <"` 8  z2 , <"`` 8 z2 - <"` 8 c ZB . s *D T z2 / <"`]pz2 0 <"`pjz2 1 <"` p z2 2 <"`V p z2 3 <"`pYz2 4 <"`g8  z2 5 <"`8 t z2 6 <"` 8  z2 7 <"`` 8 z2 8 <"` 8 c ZB 9 s *DIl 9 z2 : <"`gpz2 ; <"`ptz2 < <"` p z2 = <"`` p z2 > <"` pcz2 ? <"`q8  z2 @ <"`$8 ~ z2 A <"` 8  z2 B <"`j 8 z2 C <"`8 m ZB D s *Df  E 0. Fs  8Tree 4 2H  0޽h ? 3333___PPT10i.Ow!+D=' B= @B +$  0 ##EF:#(  x  c $ؽ.`P  . x  c $. . b!8 ]}m} F}]m}z2  <"`]qz2  <"`qjz2  <"` q z2  <"`V q z2   <"`qYz2   <"`^pz2   <"`pkz2   <"` p z2   <"`W p z2  <"`pZ  C B CDEFԔP  @  "`}-p  C BQCDEFԔb1{^WNQ@  "` -z2  <"`g9  z2  <"`9 t z2  <"` 9  z2  <"`` 9 z2  <"` 9 c `B  0DԔ f f   C BCDEFԔPu+e@  "`jf }`B  0DԔ--9 z2  <"`]pz2  <"`pjz2  <"` p z2  <"`V p z2  <"`pYZB  s *D 9 z2  <"`g8  z2   <"`8 t z2 ! <"` 8  z2 " <"`` 8 z2 # <"` 8 c z2 $ <"`]pz2 % <"`pjz2 & <"` p z2 ' <"`V p z2 ( <"`pYz2 ) <"`g8  z2 * <"`8 t z2 + <"` 8  z2 , <"`` 8 z2 - <"` 8 c ZB .B s *Df 9 z2 / <"`]pz2 0 <"`pjz2 1 <"` p z2 2 <"`V p z2 3 <"`pYz2 4 <"`g8  z2 5 <"`8 t z2 6 <"` 8  z2 7 <"`` 8 z2 8 <"` 8 c ZB 9 s *D 9 z2 : <"`gpz2 ; <"`ptz2 < <"` p z2 = <"`` p z2 > <"` pcz2 ? <"`q8  z2 @ <"`$8 ~ z2 A <"` 8  z2 B <"`j 8 z2 C <"`8 m ZB DB s *D?9  E 0. Fs  8Tree 5 2H  0޽h ? 3333___PPT10i.Pw0+D=' B= @B +  0   " (  x  c $.`P  . x  c $. .   0@.2  SThese trees are isomorphic to : 2 l      ,$D0t2  6"`Dt2  6"`C s t2  6"`  ` s t2  6"`  " s t2   6"` s t2   6"` s t2   6"`  t2   6"` " t2   6"` ` t2  6"`C  TB  c $Dpp TB B c $D3 p TB B c $D p TB  c $Dp TB  c $Dp TB  c $D3 s 3 TB  c $D3 s p TB  c $D s TB B c $D s   0.\pX< ?Done! 2,H  0޽h ? 3333___PPT10n.Pw+9pDB' B= @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+  0 @<0(  <x < c $.  . x < c $.x  . H < 0޽h ? 3333___PPT10i.p+D=' B= @B +  0 L(  Lx L c $1M 6  1  L H1   1 RB L s *Dv8 ozzH L 0޽h ? ff>>\`Y{ff___PPT10i.@=#z+D=' B= @B +  0 P(  Px P c $1   1  P H1 )  1 RB P s *Dv8 ozzH P 0޽h ? ff>>\`Y{ff___PPT10i.@=#z+D=' B= @B +   0 $T(  T~ T s *$1L 1 RB T s *Dv8 ozz T H,%1   1 H T 0޽h ? ff>>\`Y{ff___PPT10i.ͅp+D=' B= @B +  0 `&'2(  RB  s *Dv8 ozz  <,1  ^ 1   6-1  4MHCP@2  <21J_ B 2  <01M _ B 2  <91  _ B 2   <=16 J  B 2   <d@16 M  B 2   <E16   B 2   <PI1g J  B 2   < L1g M  B 2  <@Q1g   B   0>\`Y{ff___PPT10i.p@u+D=' B= @B +  0 p(  RB  s *Dv8 ozz  <1   1   6T1  4MHCP@2  <1J_ B 2  <d1M _ B 2  <1  _ B 2   <P16 J  B 2   < 16 M  B 2   <@16   B 2   <1g J  B 2   <1g M  B 2  <1g   B   01c   3 1 2 V022&&&&8X  <л1"`o4  Dc FB  S D~ FB  S D~ FB  S DOo4OFB  S Do4^B @ 6D8c LB  c $D12z c RB  s *D12{ c RB  s *D12 c   01@  ?1" 2g  01M q  ?2" 2g  01 ?3" 2gH  0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +  0 &' 2(   RB   s *Dv8 ozz   <1   1    6J  4MHCP@2   <01J_ B 2   <1M _ B 2   <<1  _ B 2   <16 J  B 2   <16 M  B 2   <16   B 2   <1g J  B 2   <X1g M  B 2   <3g   B    0l1 c  i4 6 5 6 5 4 5 4 6 022&8X   <3"` o4  Dc FB   S D  FB   S D  FB   S D o4 FB   S D o4 ^B  @ 6D8c  LB   c $D1u y LB   c $D1x z z | LB   c $D1{  LB   c $D1c z LB   c $D1c { LB   c $D1c  LB   c $D1c z LB   c $D1c { LB   c $D1c     0|3 @  ?4" 2g   03 ?  ?4" 2g !  03 J  ?4" 2g "  0#3 M  ?5" 2g #  00'3 ]  ?5" 2g $  0(+3   ?5" 2g %  0.3   ?6" 2g &  023   ?6" 2g '  063 `  ?6" 2gH   0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +  0 &'(  RB  s *Dv8 ozz  <A3   3   6B3  4MHCP@2  < G3J_ B 2  <E3M _ B 2  <N3  _ B 2   <R36 J  B 2   <pU36 M  B 2   <Z36   B 2   <\^3g J  B 2   <a3g M  B 2  <Lf3g   B   0Hi3 c i7 9 8 9 8 7 8 7 9 022&8X  <s3"` o4 Dc FB  S D FB  S D FB  S DUo4UFB  S Do4^B @ 6D8c   0w3QK%  ?7" 2g  0{3C   ?8" 2g  03CP    ?9" 2gLB  c $D12z LB  c $D12{ LB  c $D12z LB  c $D12{   # B!CaDEF1!!1AQ!a@  c"$` 2   # B!CaDEF1!!1AQ!a@  c"$` 2z   # B!CaDEF1!!1AQ!a@  c"$` 2P    # BCaDEF1>|"n~a@  c"$` 2  ! # BCaDEF1a "n@  c"$` 2  " 03m # q  ?7" 2g # 03U m # )  ?7" 2g $ 0@~3z 0  ?8" 2g % 0L3 z 0  ?8" 2g & 03Z  ?9" 2g ' 0 3 g  ?9" 2gH  0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +Z  0 qi&'(  RB  s *Dv8 ozz  <3   3   6 3  4MHCP@2  <t3J_ B 2  <KM _ B 2  <@3  _ B 2   <836 J  B 2   <36 M  B 2   <D36   B 2   <3g J  B 2   <3g M  B 2  <D3g   B LB  c $D12z LB  c $D12{   # BCaDEF1>|"n~a@  c"$` 2   03m # q  ?7" 2g  0L3 z 0  ?8" 2g  03 g  ?9" 2g  03] 2 hC1J 2GOGLB  c $D1u y LB  c $D1x z z | LB  c $D1{    03 J  ?4" 2g  0L3   ?5" 2g  03 `  ?6" 2gLB  c $D1DH LB  c $D1Gz z K LB  c $D1J N   03J ?1" 2g   03  ?2" 2g ! 03 `  ?3" 2g " 03S  `Bipartite difference = 0, 2t5t5 # 03 V  `Bipartite difference = 0, 2t5t5 $ 03X, `Bipartite difference = 1, 2t5t5^B %@ 6DjJ` ^B &@ 6DjJ `  ^B '@ 6DjJ` H  0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +Z  0 qi&'(  RB  s *Dv8 ozz  < 5   5   6 5  4MHCP@2  <KJ_ B 2  <5M _ B 2  <5  _ B 2   <56 J  B 2   <t56 M  B 2   <!56   B 2   <`%5g J  B 2   <)5g M  B 2  <L-5g   B   005] 2 hC2J 2GOG  0P55S  `Bipartite difference = 1, 2t5t5  095 V  `Bipartite difference = 1, 2t5t5  005X, `Bipartite difference = 2, 2t5t5^B @ 6DjJ` ^B @ 6DjJ `  ^B @ 6DjJ` LB  c $D12z c LB  c $D12{ c LB  c $D12 c   08D5@  ?1" 2g  0|H5M q  ?2" 2g  0LL5 ?3" 2gLB  c $D1c z LB  c $D1c { LB  c $D1c    0Q5 @  ?4" 2g   08U5 M  ?5" 2g ! 00Y5   ?6" 2gLB " c $D12z LB # c $D12{  $ # BCaDEF1a "n@  c"$` 2  % 0]5U m # )  ?7" 2g & 0a5z 0  ?8" 2g ' 0Te5Z  ?9" 2gH  0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +  0 &'U(  RB  s *Dv8 ozz  <r5   5   6r5  4MHCP@2  <Dw5J_ B 2  <v5M _ B 2  < 5  _ B 2   <؂56 J  B 2   <56 M  B 2   <Ȋ56   B 2   <5g J  B 2   <<5g M  B 2  <p5g   B   0l5] 2 hC3J 2GOG  0|5S  `Bipartite difference = 2, 2t5t5  05 V  `Bipartite difference = 2, 2t5t5  0|5X, `Bipartite difference = 0, 2t5t5^B @ 6DjJ` ^B @ 6DjJ `  ^B @ 6DjJ` LB  c $D12 c LB  c $D12z c LB  c $D12{ c   05?  ?1" 2g  05 ] q  ?2" 2g  05  ?3" 2gLB  c $D1c  LB  c $D1c z LB  c $D1c {   05 ?  ?4" 2g   0ܽ5 ]  ?5" 2g ! 05   ?6" 2g " 05QK%  ?7" 2g # 0T5C   ?8" 2g $ 05CP    ?9" 2g % # B!CaDEF1!!1AQ!a@  c"$` 2  & # B!CaDEF1!!1AQ!a@  c"$` 2z  ' # B!CaDEF1!!1AQ!a@  c"$` 2P H  0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +  0 0( (   RB   s *Dv8 ozz   B5   5    <5  %  5 H   0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +  0  !$w(  $RB $ s *Dv8 ozz $ <X5   5  $ 6\5  4MHCP@2 $ <5J_ B 2 $ <<5M _ B 2 $ <8G5  _ B 2  $ <56 J  B 2  $ <L56 M  B 2  $ <56   B 2  $ <5g J  B 2  $ <6g M  B 2 $ <|6g   B LB $ c $D12z LB $ c $D12{  $ # BCaDEF1>|"n~a@  c"$` 2  $ 0< 6m # q  ?7" 2g $ 06 z 0  ?8" 2g $ 06 g  ?9" 2g $ 06] 2 hC1J 2GOGLB $ c $D1u y LB $ c $D1x z z | LB $ c $D1{   $ 05 J  ?4" 2g $ 0$6   ?5" 2g $ 0h"6 `  ?6" 2gLB $ c $D1DH LB $ c $D1Gz z K LB $ c $D1J N  $ 0'6J ?1" 2g  $ 0X+6  ?2" 2g !$ 0(/6 `  ?3" 2gH $ 0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +"  0 !!99(k!(  (RB ( s *Dv8 ozz ( 0p>6 ~ @ 2  ( <H?6   6  ( 6\A6  4MHCP@2 ( <E6J_ B 2 ( <D6M _ B 2 ( <M6  _ B 2  ( <DQ66 J  B 2  ( <T66 M  B 2  ( <4Y66   B 2  ( <\6g J  B 2  ( <_6g M  B 2 ( <d6g   B LB ( c $D12z LB ( c $D12{  ( # BCaDEF1>|"n~a@  c"$` 2  ( 0h6m # q  ?7" 2g ( 0l6 z 0  ?8" 2g ( 0Tp6 g  ?9" 2g ( 0Dt6] 2 hC1J 2GOGLB ( c $D1u y LB ( c $D1x z z | LB ( c $D1{   ( 0y6 J  ?4" 2g ( 0 ~6   ?5" 2g ( 06 `  ?6" 2gLB ( c $D1DH LB ( c $D1Gz z K LB ( c $D1J N  ( 0Ѕ6J ?1" 2g  ( 06  ?2" 2g !( 06 `  ?3" 2gLB "( c $D12z 2LB #( c $D12{ 2LB $( c $D1c z c LB %( c $D1c { c LB &( c $D1 z LB '( c $D1 {  (( # BCDEF1D>]|D@  c"$`  2 )( # BCDEF1D>]|D@  c"$`   c  *( # BCDEF1D>]|D@  c"$` 9   +( 064  ?4" 2g ,( 06Z ?5" 2g -( 0М6 m  ?6" 2g .( 0Ġ6 k  ?7" 2g /( 0p6 M  T  ?8" 2g 0( 0h6 ]   ?9" 2g 1( 0<69 k  ?1" 2g 2( 069 m  ?2" 2g 3( 0D6   ?3" 2g 4( 0\6& [Color set = {4,5,6}, 2t5t5 5( 06 &  [Color set = {7,8,9}, 2t5t5 6( 06 &  [Color set = {1,2,3}, 2t5t5^B 7(@ 6DjJ2 2^B 8(@ 6DjJc c ^B 9(@ 6DjJ  H ( 0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +  0  !,{(  ,RB , s *Dv8 ozz , <6   6  , 6l$K  4MHCP@2 , <6J_ B 2 , <6M _ B 2 , <6  _ B 2  , <66 J  B 2  , <t66 M  B 2  , <66   B 2  , <`6g J  B 2  , <6g M  B 2 , <P6g   B  , # BCaDEF1>|"n~a@  c"$` 2  , 06 z 0  ?8" 2g , 06] 2 lC1 J 2GOGLB , c $D1u y  , 07 J  ?4" 2gLB , c $D1J N  , 07 `  ?3" 2gLB , c $D12z 2LB , c $D12{ 2LB , c $D1c z c LB , c $D1c { c LB , c $D1 z LB , c $D1 {  , 0X 7Z ?5" 2g , 06 m  ?6" 2g , 07 k  ?7" 2g , 0`7 ]   ?9" 2g  , 079 k  ?1" 2g !, 0|79 m  ?2" 2gH , 0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +(  0 ?7 !0(  0RB 0 s *Dv8 ozz 0 <'7   7  0 6)7  4MHCP@2 0 <,7J_ B 2 0 <07M _ B 2 0 <47  _ B 2  0 <876 J  B 2  0 <p;76 M  B 2  0 <@76   B 2  0 <\D7g J  B 2  0 <G7g M  B 2 0 <LL7g   B LB 0 c $D12z LB 0 c $D12{  0 0O7m # q  ?7" 2g 0 0S7 g  ?9" 2g 0 04W7] 2 lC1 J 2GOGLB 0 c $D1x z z | LB 0 c $D1{   0 00]7   ?5" 2g 0 0 a7 `  ?6" 2gLB 0 c $D1DH LB 0 c $D1Gz z K  0 0\e7J ?1" 2g 0 0h7  ?2" 2g 0 # BCDEF1D>]|D@  c"$`  2 0 # BCDEF1D>]|D@  c"$`   c  0 # BCDEF1D>]|D@  c"$` 9   0 0m74  ?4" 2g  0 0q7 M  T  ?8" 2g !0 0t7   ?3" 2gH 0 0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +  0 `DN(  Dx D c $d7   7  D B 7 `  7 H D 0޽h ? ff>>\`Y{ff___PPT10i.p@u+D=' B= @B +  0   0(   x   c $7`P  7 x   c $@7  7 H   0޽h ? 3333___PPT10e.+D=' B= @B +  0 X<(  X~ X s *7@(   7 ~ X s *7  7 H X 0޽h ? 3333___PPT10i.'q+D=' B= @B +  0 SKP(  ~  s *L7`P  7 ~  s *ԕ76$  7 F  07Zz d is a (2m-1)-edge-coloring of K2m, and (xy) = c.D3 2  ( z  .  . ,$D0n2  0"`9 9 n2  0"` g6 n2  0"` 6 n2   0"`cn2   0"`vcn2   0"`3 9 TB  B c $D` p6 TB  B c $D` p6 TB  c $D!p!TB  c $D p9 TB  c $Dpf   07. 3u 2  0795  3v 2  0ܳ7Cl&S  71 2  047p S  72 2  07    73 2  07   74 2  07sV  75 2  07G Y.,$0 n= u-1 (4)> 2  07F ,$0 PSu2 2 z       ,$D0ZB B s *D    07   du9 4: or v9 4: 0 2H  0޽h ? 338 0 ___PPT10 .t*#+ҞCDt ' B= @B D/ ' = @BA?%,( < +O%,( < +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<*D' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*D' =%(D' =%(DX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*+p+0+0 ++0+0 +'  0 `33(  x  c $7MDY( 7   s 70e0e #" 0e`P  7 z * E   * E,$D0ZB  s *D* w ZB B s *D   07  73 2  07^ yE 74 2 z F    F ,$D0 N 1 ]   F rn2   0"` ; n2   0"` d n2   0"` F n2  0"`I1  n2  0"`In2  0"` m ZB B s *D8c}  ZB B s *D8c}  ZB  s *D8c} ZB  s *D8cr} ; ZB  s *D8c}    07vZ] 3u 2  0: ~ e 3v 2  0X:z a  71 2  0: h O  72 2  0 :r' 8  73 2  0:m T  74 2  0: u  75 2  0: 5T 2m z 8 P  8 P,$D0n2  0"`-f  n2   0"`Vl  n2 ! 0"`8 l n2 " 0"`)n2 # 0"`p ) n2 $ 0"`_ f ZB %B s *D8c a ZB &B s *D8c a ZB ' s *Da ZB ( s *D8ca ZB ) s *Dag  * 0:Z A 3u 2 + 0 !:b PI  3v 2 , 0@$:M^E  71 2 - 0":uL ;3  72 2 . 0x&:k  75 2ZB / s *D8c  ZB 0B s *D8c[ w  1 0/:{ A  73 2 2 03:-  74 2 3 06: T[u,v]D 2H  0޽h ? 33  ___PPT10 .~p + D~ ' B= @B D9 ' = @BA?%,( < +O%,( < +D' =%(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<*%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*+  0 p<(  ~  s *LS:`P  : ~  s *S: : H  0޽h ? 33___PPT10i.Ҩ+D=' B= @B +W  0 nf (  ~  s *(x:  :   s *~: : "p`Pp 0 N ? ?  :  0L bZT = {T1,& ,Tn}, xi is the root of Ti. Let U = . 2C     " 0  `A ? ?]  :x  <A ??F ~  BA $? ? } )  $~   BA ? ? H  0޽h ? 33___PPT10i.d+D=' B= @B +  0 CC(  ~  s *L`P  L f2  0"`mf2  0"`mkf2  0"`m0 f2  0"`Vaf2  0"`C9f2  0"`8f2   0"`m Q f2   0"`m r2   <"`F& f2   0"`E t f2   0"`E : f2  0"`E J f2  0"`E  f2  0"`E { f2  0"`S 0 f2  0"`S f2  0"`S  a f2  0"`S p f2  0"`S 6 f2  0"`S t f2  0"`S  f2  0"`S * f2  0"`S  f2  0"` Z f2  0"`  f2  0"` j f2  0"`s  3 f2  0"`r f2  0"`r C f2   0"`F  f2 ! 0"`F 6 f2 " 0"`F t  # 0uLf Fx1( 2 $ 0xL=z  Fx2( 2 % 0<~L @  Fx3( 2 & 0|:   Fx4( 2 ' 0@:  Fx5( 2LB (@ c $D LB )@ c $DW LB *@ c $D LB + c $D] LB , c $D LB - c $D3 LB . c $D LB / c $D  LB 0 c $DF LB 1@ c $Ds LB 2@ c $D s LB 3@ c $Dws LB 4 c $Ds LB 5 c $DM s LB 6 c $D s LB 7 c $Dcs LB 8 c $Ds LB 9 c $D] ] LB : c $D LB ;@ c $Dws LB <@ c $DM # s LB = c $D] 3 LB > c $D# cs LB ?@ c $D LB @ c $D S LB A c $D F r2 B <"`'   C 0:   5U 2H  0޽h ? 33___PPT10i.d+D=' B= @B +  0 55((  ~  s *|:`P  : f2  0"`mf2  0"`mkf2  0"`m0 f2  0"`Vaf2  0"`C9f2  0"`8f2   0"`m Q f2   0"`m r2   <"`F& f2   0"`E t f2   0"`E : f2  0"`E J f2  0"`E  f2  0"`E { f2  0"`S 0 f2  0"`S f2  0"`S  a f2  0"`S p f2  0"`S 6 f2  0"`S t f2  0"`S  f2  0"`S * f2  0"`S  f2  0"` Z f2  0"`  f2  0"` j f2  0"`s  3 f2  0"`r f2  0"`r C f2   0"`F  f2 ! 0"`F 6 f2 " 0"`F t  # 0D:f Fx1( 2 $ 0:=z  Fx2( 2 % 04: @  Fx3( 2 & 0:   Fx4( 2 ' 0$:  Fx5( 2Wz  c  (  c ,$D0 ) 0D? | c  ;u 2"N   *  TB + c $D] TB , c $D] TB - c $D] ] TB .B c $D] # TB /B c $D]  z -P 7 0 P- 7,$D0 1 0?-P7 ;1 2 2 0 ?jPL 7 ;2 2 3 0: P 7 ;3 2 4 0? P 7 ;4 2 5 0?. P 7 ;5 2H  0޽h ? 33___PPT10.d+գ<D~' B= @B D9' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(+3c  0 44ffu4(  ~  s *:`P  ? f2  0"`mf2  0"`mkf2  0"`m0 f2  0"`Vaf2  0"`C9f2  0"`8f2   0"`m Q f2   0"`m r2   <"`F& f2   0"`E t f2   0"`E : f2  0"`E J f2  0"`E  f2  0"`E { f2  0"`S 0 f2  0"`S f2  0"`S  a f2  0"`S p f2  0"`S 6 f2  0"`S t f2  0"`S  f2  0"`S * f2  0"`S  f2  0"` Z f2  0"`  f2  0"` j f2  0"`\  f2  0"`[ U f2  0"`F  f2   0"`F  f2 ! 0"`F 7  " 0%?f Fx1( 2 # 0+?=z  Fx2( 2 $ 00? @  Fx3( 2 % 0X6?   Fx4( 2 & 0;?  Fx5( 2 ' 0HA?| c  ;u 2Fz  g  (  g ,$D0TB ) c $D]  * 0dF?-P7 ;1 2 + 0`K?n g  Fv1( 2TB , c $D  - 0TP?(  h  ;6 2Fz  ^  .   ^ ,$D0 / 0T?w 0 ^  Fu1( 2TB 0B c $DM ] F TB 1B c $D s  2 0Z? ? &  ;1 2 3 0_?1   ;6 2Fz jc  4 jc ,$D0TB 5 c $D]  6 0d?jPL 7 ;2 2 7 0i? | c  Fv2( 2TB 8 c $D3  9 0tI?# > %  ;7 2f2 : 0"`F " } f2 ; 0"`F ` f2 < 0"`F ) Fz  ` ^  =  ` ^ ,$D0 > 0s?w ^  Fu2( 2TB ?B c $D ] s  @ 0x?=   ;7 2TB AB c $DP 3 s  B 0}?~ _ ` F  ;2 2Fz  g  C  g ,$D 0TB D c $D] ]  E 0̂? P 7 ;3 2 F 0ć?D g  Fv3( 2TB G c $D]   H 0? \C  ;8 2Fz  ^  I  ^ ,$D 0 J 0l?w Z^  Fu3( 2TB K c $Dw ] s  L 0 ?6_ F  ;8 2TB MB c $D s  N 0? _ F  ;3 2Fz ] g  O ] g ,$D0TB PB c $D] #  Q 04? P 7 ;4 2 R 0,? g  Fv4( 2TB S c $D#    T 0H?>%  ;9 2Fz  Z  U   Z ,$D0 V 0ԯ?s Z  Fu4( 2TB WB c $D s TB X c $D  ] s  Y 0Ե?d K  ;9 2 Z 0?_ F  ;4 2Gz ] )g  [ ] )g ,$D0TB \B c $D]   ] 0?. P 7 ;5 2 ^ 0? )g  Fv5( 2TB _ c $D F  ` 0?>F%  <10 2Gz G F^  a  GF^ ,$D0 b 0?Gw ^  Fu5( 2TB cB c $D  Fs TB d c $D ] s  e 0?*c \J  <10 2 f 04?N_ &F  ;5 2H  0޽h ? 33--___PPT10-.d+D-' B= @B D]-' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*.%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*(%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*.%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*=%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*4%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*=%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*C%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*I%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*O%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*U%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*O%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*U%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*[%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*a%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*[%(D2' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*a%(+ E  0 ,,ZZ,(  ~  s *?`P  ? f2  0"`mf2  0"`mkf2  0"`m0 f2  0"`Vaf2  0"`C9f2  0"`8f2   0"`m Q f2   0"`m r2   <"`F& f2   0"`E t f2   0"`E : f2  0"`E J f2  0"`E  f2  0"`E { f2  0"`S 0 f2  0"`S f2  0"`S  a f2  0"`S p f2  0"`S 6 f2  0"`S t f2  0"`S  f2  0"`S * f2  0"`S  f2  0"` Z f2  0"`  f2  0"` j f2  0"`s  3 f2  0"`r f2  0"`r C f2   0"`F  f2 ! 0"`F 6 f2 " 0"`F t  # 0?s EZ  Fu5( 2 $ 0DAs Z  Fu4( 2 % 0@As ,Z  Fu3( 2 & 0 As ZZ  Fu2( 2 ' 0As  Z  Fu1( 2#F  c  (  c  ) 0A | c  ;u 2"N   *  TB + c $D] TB , c $D] TB - c $D] ] TB .B c $D] # TB /B c $D]  F -P 7 0 P- 7 1 0dA-P7 ;1 2 2 0LAjPL 7 ;2 2 3 0p"A P 7 ;3 2 4 0'A P 7 ;4 2 5 0+A. P 7 ;5 2z  ] w  6  ] w ,$D0TB 7B c $DM ] s TB 8B c $D ] s TB 9 c $Dw 0 s TB : c $D  0 s TB ; c $D 0 s  < 00AM  s  ;6 2 = 05A= s  ;7 2 > 03A- s  ;8 2 ? 0=A Jw  ;9 2 @ 0BA s  <10 2LB A c $D ] LB B c $D ] F B C c $D 3 s ,$@0z t Ws  D  tWs ,$D 0 E 0HAt )  ;6 2TB F c $D Ws z M   G  M ,$D0TB H c $DM s s  I 0NA  ]  ;7 2 J 0RA  ,$0 ;8 2z  _  K  _ ,$D0TB L c $D b_  M 00WA   ;9 2z   Zs  N  Zs ,$D0TB O c $D  Zs  P 0t\A   <10 2LB Q c $D2]  LB R c $D]  LB S@ c $D Q  T BCDEFZZ@  W]  U BqCDEFq3|>@  p]  V BC[DEFO-[[[-@   3  W BCDEF2[d[@   ] F6 LB X c $D ] cs LB Y c $D ] s  Z BC-DEF-F>b-@  \ xN H  0޽h ? 33:2___PPT10.d+YVD' B= @B Di' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*6%(D' =%(D' =%(DE' =4@BBB B%(D' =-6B'blinds(horizontal)*<3<*6D' =1:Bhidden*o3>+B#style.visibility<*6%(D' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =-6B+checkerboard(across)*<3<*GD[' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =-6B+checkerboard(across)*<3<*CDX' =A@BBB B0B%(D' =1:Bvisible*o3>+B#style.visibility<*J%(D' =-6B+checkerboard(across)*<3<*JD' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*K%(D' =-6B+checkerboard(across)*<3<*KD' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*N%(D' =-6B+checkerboard(across)*<3<*ND' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*D%(D' =-6B+checkerboard(across)*<3<*D+8+0+J0 +{   0 (  ~  s *rA8(  A :  0sAT D<4___PPT9 &Colors 6~10 are distinct. v19 7: , v29 8: , v39 9: , v49 10: , v59 6: are not incident to x1 ~ x5. v19 7: , v29 8: , v39 9: , v49 10: , v59 6: can not form a cycle.6 2            8X  0In order to get vn, |U| e" 6n-4. 2G G O G  G G H  0޽h ? 33 ___PPT10..~@+eD' A= @B DD' = @BA?%,( < +O%,( < +D{' =%(D#' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*D' =+4 8?dCB1+#ppt_h/2BCB#ppt_yB*Y3>B ppt_y<*+8+0+0 +_  0 E+=+WW*(  ~  s *A  A   s *lA A "p`Pp 0  `A  ? ? 3   Az V   V ,$@0`2  0S W `2  0 7 `2  0 7V `2   0  `2   0 V n2   0"`G  n2   0"`  n2   0"` X n2  0"`g  n2  0"`w    0A  Fx1( 2TB B c $D  * TB  c $D   TB B c $D * TB  c $D  TB B c $Dt * TB  c $D*  TB  c $D*  B  c $D n n ,$@0B  c $D n # ,$@0z  Z    Z ,$@0`2  0@ W `2  0 { `2  0 { Z `2  0 0 `2  0 0 Z n2   0"` n2 ! 0"` Q n2 " 0"`A n2 # 0"`  n2 $ 0"`   % 0LL }  HXn-1( 2TB &B c $DM n TB ' c $DM M TB (B c $D n TB ) c $D  TB *B c $D n TB + c $Dn TB , c $Dn  - 09L Z ,$0 ?. . . 2 . 0>L  ,$ 0 Fv1( 2 / 0A ,$ 0 Fv2( 2 0 0hA  ,$0 Fu1( 2 1 0A E U ,$0 Fu1( 2| z w  2 w ,$D0n2 3 0"`S ) n2 4 0"` 6 n2 5 0"` 6V TB 6 c $D  n2 7 0"`s  n2 8 0"`s V TB 9 c $D   : 0Acc tJ  =. . 2n2 ; 0"`V  n2 < 0"`  g n2 = 0"` w TB >B c $D   TB ?B c $D  TB @ c $D  TB A c $D 9 TB B c $D I  C 0Ai  ;u 2 D 0A/   Fu1( 2 E 0B%)   Hun-1( 2Zz v F  G ,$@0n2 G 0"`vun2 H 0"`vn2 I 0"`v+n2 J 0"`:v K 0B  ,$ 0 ;u 2 L 00 B  # ,$ 0 ;u 2B M c $D  ,$@ 0B N c $D g ,$@0B O c $D @ ,$@0B P c $D P ,$@0B Q c $D ** ,$@0B R c $D * ,$@0Zz v S   ,$D0n2 T 0"`vun2 U 0"`vn2 V 0"`v+n2 W 0"`:vH  0޽h ? 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