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Unicyclic Spanning Subgraphs Conclusionzlx2g)g$g2g gt  P             Preliminariesa  LA (proper) k-edge-coloring of a graph G is a mapping from E(G) into {1,& ,k } such that incident edges of G receive distinct colors. Example: 3-edge-coloring of K4 ^x  gg gggggggggggog   Preliminariesa  nA subgraph in an edge-colored graph is said to be multicolored if no two edges have the same color. Example:tox2g g3(ggc4  d   Preliminariesa  RG =A*"B is a balanced bipartite graph. Let A = { a1, a2, & , an } and B = { b1, b2, & , bn }, then the bipartite difference of aibj is defined as j  i (mod n). Example:xx xggg$ggggogggogoggggogogog ggggogoggggggggcNU  &  (  T= Preliminariesa  A k-total-coloring of a graph G is a mapping from V(G)*"E(G) into {1,& ,k } such that : (1) adjacent vertices in G receive distinct colors, (2) incident edges of G receive distinct colors, and (3) any vertex and its incident edges receive distinct colors.0xgg3 ggg ggg'gg0ggagg   Preliminariesa   Idempotent commutative latin square of order n a" n-total-colored Kn a" n-edge-colored Kn,n^xg g ggggggggoggggooogZ  %     Multicolored 1-factora  Conjecture (Transversal) In any n-edge-colored Kn,n, there exists a multicolored 1-factor for each odd integer n. Theorem (Woolbright and Fu, 1998) In any (2m-1)-edge-colored K2m where m>2, there exists a multicolored 1-factor.  x gg gggggooo<gggggg% ggggoogg+g@/  J   `  Multicolored Subgrapha&    Given an (n 1)-edge-colored Kn, a multicolored Hamiltonian path exists for n > 4. (!?) How about multicolored spanning trees?x gggggogg gg"ggg&  a ! Known Resultsa  ,Brualdi-Hollingsworth s Conjecture (1998) If m>2, then in any proper (2m 1)-edge-coloring of K2m, all edges can be partitioned into m multicolored spanning trees. Theorem (Krussel, Marshall and Verall, 2000) Three edge-disjoint multicolored spanning trees always exist.x"gge%ggggggoo$gggg$g%gg=gf B   d    M U>Multicolored Tree Parallelism ve aeae a   V?Multicolored Tree Parallelism ve aeae a  Definition 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. x gggoo gggggggggoo-ggg gg  " Conjecture a  Constantine s Weak Conjecture (2002) For any m>2, K2m can be (2m 1)-edge-colored in such a way that the edges can be partitioned into m multicolored isomorphic spanning trees. Constantine s Strong Conjecture (2002) If m>2, then in any proper (2m 1)-edge-coloring of K2m, all edges can be partitioned into m multicolored isomorphic spanning trees. ^xggg% ggggoo ggEggg ggggg%gg gg gg  ggoo$ggg ggJA      d  hG,Multicolored Hamiltonian Cycle Parallelism -g$ c$g$ g$g$g$c$g$ c$ - Definition K2m 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. x gggoo g*ggggggggooo-gg#g  # Known Resultsa  PConstantine s Weak Conjecture (2005) For any natural number m, K2m+1 admits an MHCP. Theorem (Constantine, 2005) If 2m+1 is an odd prime, then K2m+1 admits an MHCP.xggg%ggggooogggg%gg ggggooog  YA Conjecture a  Constantine s Strong Conjecture (2005) In any proper (2m+1)-edge-coloring of K2m+1, all edges can be partitioned into m multicolored isomorphic spanning unicyclic subgraphs. Note. Unicyclic: a graph with exactly one cycle.x2xggg%gg ggcggooo$ggg g gg5gZ          " $ Known Resultsa  fF Known Resultsa   !Multicolored Subgraph Parallelism""a&     1. Existence of MTP"g(3   MTPa  HConstatine s Weak Conjecture (2002) For any m>2, K2m can be (2m 1)-edge-colored in such a way that the edges can be partitioned into m multicolored isomorphic spanning trees. Theorem 2.1 (Constantine, 2002) If m`"1 or 3 and K2m admits an MTP, then for all r 1, admits an MTP.lx lSggg% ggggoo ggEggg ggg ggg%gg ggooggg@  4      MTPa  zTheorem 2.2 (Fu and Lo) For m`"2, K2m admits an MTP. Pf: > gg gggggoog > !Multicolored Subgraph Parallelism""a&     2. Existence of MHCP"g(3   MHCPa  Def: 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. Theorem 2.3 (Constantine, 2005) If n is an odd prime, then Kn admits an MHCP. lxlgggooo g*ggggggggooo-gggg3g ggg%ggggogg4      MHCPa  Lemma 2.4 (Fu and Lo) Let v be a composite odd integer and p be the smallest prime with p|v. Assume v = mp. If Km admits an MHCP, then Km(p) has a proper mp-edge-coloring that admits an MHCP."x g ggg ggggggggggoggoooogg#gcN[  .  4   MHCPa  PTheorem 2.5 (Fu and Lo) For each odd integer v 3, Kv admits an MHCP. Pf: Q eg egeeeemegeg&6    MHCPa  XExample: K15 Start from a 5-total-colored K5. (Consider K5(3) first.) Yx ggo$ggog o ggo g Y MHCPa   Example: K15`  ggog  (MHCPa   Example: K15`  ggog  )MHCPa   Example: K15`  ggog  * MHCPa   Example: K15`  ggog  +!MHCPa   Example: K15`  ggog  ,"MHCPa   Example: K15`  ggog  W@MHCPa  Example: K15 bx ggog  'MHCPa   Example: K15P  cckc  -#MHCPa   Example: K15P  cckc  /%MHCPa   Example: K15P  cckc  .$MHCPa   Example: K15P  cckc  0&#Existence of Multicolored Subgraphs$$c$(    1. Spanning Trees"g(3  1'MSTa  Definition  is a (2m 1)-edge-coloring of K2m, and T is a multicolored spanning tree with root x. If x is incident to two leaves e1 = xy1 and e2 = xy2, then let T[x;y1,y2;z1,z2] = T  e1  e2 + y1z1 + y2z2, for some z1, z2. eeeeeeeeeemmee+eeeeeemeemeemeem eeeeeemeemeemeemeeeemeemeememeemem eemeeme  7*MSTa  Example6ee  8+MSTa  Example6ee  6)MSTa  Theorem 3.1 (Fu and Lo) Let  be an arbitrary (2m 1)-edge-coloring of K2m and x0 be an arbitrary vertex. Then, there exist mutually edge-disjoint multicolored spanning trees, each of which contains a pendent vertex x0. ~t e eeggggggooggo+gagggog c   ZBMSTa  TNotation Tj(i) : the jth spanning tree which was constructed at round i. HU eeemmmmeem0eeeeg c @     : 4(Sketch proof of Thma&   @Round 1. Pick any one vertex x1. Let T1(1) = x1-star.Axeemeemmeeme A ;-Sketch proof of Thma&   Round 2. Pick x2, u.xeemeee  <.Sketch proof of Thma&    Round 2.  x e  =/Sketch proof of Thma&   )Round 2. Use u, u1 to construct 2nd tree.*x eeeememe * >0Sketch proof of Thma&    ?1Sketch proof of Thma&    @2Sketch proof of Thma&   Round 3. Pick x3, u.xeemeee  B3Sketch proof of Thma&   Round 3. Pick x3, u.xeemeee  :,Sketch proof of Thma&   ZDiscussion: (1) Adjust T1(n-1) ! T1(n), """""", Tn-1(n-1) ! Tn-1(n). (2) Define Tn(n) = xn-star  {"""} +{"""}. (3) Above works whenever |U| 9n-14. (4) |W| 2m-2n2+3n.jx e ggoooofggoooogoggooooo f ggooooo ggooooggogege" g  g  g  g  g  g  g  g  g  g  g  g  g  o  g  g  g @Q    S I5Sketch proof of Thma&   Discussion: We have n spanning trees if 2m 2n2 + 6n  14. Fixed 2m, we have at least spanning trees.  10 " d " d>0 " d egggggggogggg gg6gg } K6#Existence of Multicolored Subgraphs$$c$(    2. Unicyclic Spanning Subgraph"g(3@       L7MUSSa  xCorollary 3.2 (Fu and Lo) Letbe an arbitrary (2m 1)-edge-coloring of K2m 1. Then, there exist mutually edge-disjoint multicolored unicyclic spanning subgraphs (MUSS).e eegggggooo&ggg gg ggggg c Z  u         M8MUSSa  Proof. xe  N9  Conclusion"  g,3  kH Known Resultsa  cE Reference a   [1] S. Akbari, A. Alipour, H. L. Fu and Y. H. Lo, Multicolored parallelisms of isomorphic spanning trees, SIAM J. Discrete Math. 20 (2006), No. 3, 564-567. [2] R. A6. Brualdi and S. Hollingsworth, Multicolored trees in complete graphs, J. Combin. Theory Ser. B 68 (1996), No. 2, 310-313. [3] G. M. Constantine, Multicolored parallelisms of isomorphic spanning trees, Discrete Math. Theor. Compu. Sci. 5 (2002), No. 1, 121-125. [4] G. M. Constantine, Edge-disjoint isomorphic multicolored trees and cycles in complete graphs, SIAM J. Discrete Math. 18 (2005), No. 3, 577-580. [5] H. L. Fu and Y. H. Lo, Multicolored parallelisms of Hamiltonian cycles, Discrete Math. 309 (2009), No. 14, 4871-4876."xg      A      1 lI Reference a   [6] H. L. Fu and Y. H. Lo, Multicolored isomorphic spanning trees in complete graphs, Ars Combinatoria, to appear. [7] H. L. Fu and Y. H. Lo, Multicolored spanning trees in edge-colored complete graphs, SIAM J. Discrete Math., revised. [8] H. L. Fu and D. E. Woolbright, On the exists of rainbows in 1-factorizations of K2n, J. Combin. Des. 6 (1998), 1-20. [9] J. Krussel, S. Marshal and H. Verral, Spanning trees orthogonal to one-factorizations of K2n, Ars Combin. 57 (2000), 77-82. [10] H. J. 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C7"IDqIi+z2O$  ___PPT10 . ]+M?D` ' m= @B D ' = @BA?%,( < +O%,( < +D' =%(Du' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*#`%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*$`%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*M`%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*L`%(+p+0+$`0 ++0+M`0 +ި ! 0 e(  d d <  ,YS    d s *C<$D 0   d Z19=M n Introduction$ 2 g    d Z1  X  Va"" 2g  ql  6   e f ,$D 0B d TD1 6  d ZT1C"? s m  S1" 2g  B d TD13 6 3  d Zl1C"?~ s ` m  S2" 2g  B d TD16   d Z@1C"?s m  S3" 2g  B d TD1d6 d  d Z01C"?s m  S4" 2g  B d TD1W6 W  d Z\1C"?s m  S5" 2g  %z Gs  d# #"&S 1% d c _ж_ж1 ? i5"g   @`% d c T_ж_ж1 ?& i2"g   @`% d c l_ж_ж1 ?1& i4"g   @`% d c _ж_ж1 ?<1 i1"g   @`% d c D _ж_ж1 ?G< i3"g   @`% d c T_ж_ж1 ?( i2"g   @`% d c (_ж_ж1 ?&( i4"g   @`% d c $_ж_ж1 ?1(& i1"g   @`% d c ._ж_ж1 ?<(1 i3"g   @`% d c L7_ж_ж1 ?G(< i5"g   @`% d c (_ж_ж1 ?A ( i4"g   @`% d c I_ж_ж1 ?&A ( i1"g   @`% d c @R_ж_ж1 ?1A &( i3"g   @`% d c M_ж_ж1 ?<A 1( i5"g   @`% d c d_ж_ж1 ?GA <( i2"g   @`% d c n_ж_ж1 ?Z A  i1"g   @`% d c dv_ж_ж1 ?&Z A  i3"g   @`% d c q_ж_ж1 ?1Z &A  i5"g   @`% d c _ж_ж1 ?<Z 1A  i2"g   @`% d c <_ж_ж1 ?GZ <A  i4"g   @`% d c ؚ_ж_ж1 ?s Z  i3"g   @`% d c l_ж_ж1 ?&s Z  i5"g   @`% d c _ж_ж1 ?1s &Z  i2"g   @`% d c _ж_ж1 ?<s 1Z  i4"g   @`% d c _ж_ж1 ?Gs <Z  i1"g   @`B d C x_ж_ж1 ?Gs s B d 3 r_ж_ж1 ?GZ Z B d 3 r_ж_ж1 ?GA A B d 3 r_ж_ж1 ?G((B d 3 r_ж_ж1 ?GB d C x_ж_ж1 ?GB d C x_ж_ж1 ?Gs GB d 3 r_ж_ж1 ?<s <B d 3 r_ж_ж1 ?1s 1B d 3 r_ж_ж1 ?&s &B d 3 r_ж_ж1 ?s B d C x_ж_ж1 ?s S+    e #"&S 1,$D 0' d c _ж_ж1 ?   k5$g   @` d c _ж_ж1 ?   b$g   @` d c _ж_ж1 ?   b$g   @` d c `_ж_ж1 ?   b$g   @` d c  _ж_ж1 ?   b$g   @` d c T_ж_ж1 ?  b$g   @`' d c _ж_ж1 ?   k4$g   @` d c |_ж_ж1 ?   b$g   @` d c _ж_ж1 ?   b$g   @` d c L_ж_ж1 ?   b$g   @` d c  _ж_ж1 ?  b$g   @` d c $0_ж_ж1 ?   b$g   @`' d c  9_ж_ж1 ?   k3$g   @` d c ,B_ж_ж1 ?   b$g   @` d c K_ж_ж1 ?   b$g   @` d c S_ж_ж1 ?  b$g   @` d c \_ж_ж1 ?   b$g   @` d c (f_ж_ж1 ?   b$g   @`' d c to_ж_ж1 ?   k2$g   @` d c px_ж_ж1 ?   b$g   @` d c i_ж_ж1 ?  b$g   @` d c _ж_ж1 ?   b$g   @` e c _ж_ж1 ?   b$g   @` e c x_ж_ж1 ?   b$g   @`' e c d_ж_ж1 ?   k1$g   @`B e C x_ж_ж1 ?  B e C x_ж_ж1 ? B  e C x_ж_ж1 ?  B e C x_ж_ж1 ?B ^e C x_ж_ж1 ?  B _e C x_ж_ж1 ?  B `e C x_ж_ж1 ?  B be C x_ж_ж1 ?  B de C x_ж_ж1 ? B ge C x_ж_ж1 ?B ie C x_ж_ж1 ?  B qe C x_ж_ж1 ?B se C x_ж_ж1 ?  B {e C x_ж_ж1 ?B }e C x_ж_ж1 ?  B e C x_ж_ж1 ? B e C x_ж_ж1 ? B e C x_ж_ж1 ? B e C x_ж_ж1 ? B e C x_ж_ж1 ?  S+   e #"&S 1,$@ 0 e c _ж_ж1 ?  b$g   @` e c x_ж_ж1 ?  b$g   @` e c _ж_ж1 ?  b$g   @` e c _ж_ж1 ?  b$g   @`' e c h_ж_ж1 ?  k3$g   @`' e c l_ж_ж1 ? k2$g   @` e c _ж_ж1 ? b$g   @` e c _ж_ж1 ? b$g   @` e c _ж_ж1 ? b$g   @` e c _ж_ж1 ? b$g   @` e c _ж_ж1 ? b$g   @`' e c P_ж_ж1 ? k1$g   @` e c h!_ж_ж1 ? b$g   @` e c L*_ж_ж1 ? b$g   @` e c 03_ж_ж1 ? b$g   @` e c <_ж_ж1 ? b$g   @`  e c D_ж_ж1 ? b$g   @`' !e c M_ж_ж1 ? k5$g   @` "e c 8W_ж_ж1 ? b$g   @` #e c `_ж_ж1 ? b$g   @` $e c R_ж_ж1 ? b$g   @` %e c ,r_ж_ж1 ? b$g   @` &e c T{_ж_ж1 ? b$g   @`' 'e c _ж_ж1 ? k4$g   @` (e c ,_ж_ж1 ? b$g   @`B )e C x_ж_ж1 ?B .e C x_ж_ж1 ?  B /e C x_ж_ж1 ?B 4e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ?B e C x_ж_ж1 ? B e C x_ж_ж1 ? B e C x_ж_ж1 ?  B e C x_ж_ж1 ?  B e C x_ж_ж1 ?  B e C x_ж_ж1 ?  l  6 V  e6 V ,$D  0B e TD1 6 3 B e TD13 6  B e TD16 d B e TD1c6 V B e TD1 6 V  e Z$1 F } @  S4" 2g   e Zl14 F @  S5" 2g   e Z1T 6  S1" 2g   e Zġ1   S2" 2g   e ZH1p R  S3" 2g  l m    e m  ,$D  0n2 d 0"`m a n2 d 0"`a  n2 d 0"` a [ n2 d 0"`Ka  n2 d 0"`#a  n2 d 0"`m  h n2 d 0"` h n2 d 0"`  [ h n2 d 0"`K h n2 d 0"`# h H d 0޽h ? C7"IDqIi+z2O___PPT10m. ]+0DA' m= @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*dH^%(D' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =-6B#strips(downLeft)*<3<*eD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =%(Dd' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*e%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*e%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*e%(+ ! 0 pLB(  Lx L c $Y   L 6j ,3 S    L ZB19=M n Introduction$ 2 g   H L 0޽h ? C7"IDqIi+z2O___PPT10i. ]+D=' m= @B + ! 0 ln(  l l c $pY <$D  0   l 6H ,S    l Z19=M n Introduction$ 2 g   H l 0޽h ? C7"IDqIi+z2OIA___PPT10!. ]+o!D' m= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*lX%(D' =-s6Bwipe(down)*<3<*lX+ ! 0 pn(  p p c $LY<$D 0   p 6 , S    p Z\19=M n Introduction$ 2 g   H p 0޽h ? C7"IDqIi+z2O___PPT10n. ]+ CDB' m= @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*p%(+ ! 0  l(  l l 6  ,S    l ZH19=M n Introduction$ 2 g    l Zx1}7 Z W T1 T2 T3 Color 1 x3x5 x4x6 x1x2 Color 2 x2x4 x1x5 x3x6 Color 3 x2x5 x3x4 x1x6 Color 4 x2x6 x1x3 x4x5 Color 5 x1x4 x2x3 x5x6h 2eemeemeem eememeemememem eememeememeemem eememeememeemem eememeememeemem eememeememeemem  * l ` " l` ,$D 0t2 l 6"` ;h t2  l 6"`C  t2 l 6"`-;t2 l 6"`9  t2 l 6"`  t2 l 6"`t  B l TD1 ` 6 B lB TD1p` % B l TD1 2S B l TD1p% p  l ZlR1. l  hx16 2em   l ZU1"B hx26 2em   l ZZ1 &  hx36 2em  B l TD1;   l Z^1sCc  hx46 2em   l ZLb1` `  hx56 2em   l Zf1& F  hx66 2em   l Zk1hd F lT1: 2g o   1  l Zo1]  Ex. 5-edge-colored K6b 2ggog  H l 0޽h ? C7"IDqIi+z2O___PPT10n. 譭+Y DB' m= @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*l%(+ ! 0 pN(  p p 6y ,S    p Zz19=M n Introduction$ 2 g    p 0$~ m   H p 0޽h ? C7"IDqIi+z2O___PPT10i. 譭+D=' m= @B + ! 0 tB(  tx t c $ Yi   t 6Ѝ , S    t Z19=M n Introduction$ 2 g   H t 0޽h ? C7"IDqIi+z2O___PPT10i. ]+D=' m= @B + ! 0 N(    6Qt ,S  t   Zt19=M n Introduction$ 2 g     0n:  t H  0޽h ? C7"IDqIi+z2O___PPT10i. 譭+D=' m= @B + ! 0  xn(  x x c $i<$D 0   x 6x , S    x ZĢ19=M n Introduction$ 2 g   H x 0޽h ? C7"IDqIi+z2O___PPT10n. ]+ CDB' m= @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*xX%(+ ! 0 @n(    c $<$D 0    6\ , S     Z19=M n Introduction$ 2 g   H  0޽h ? C7"IDqIi+z2O___PPT10n. ]+ CDB' m= @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+ ! 0 @@(    <  ,YS     Z19=M n Introduction$ 2 g   p Y;  #"]Y; #  c x_ж_ж1?AF Y; g? g0   @`#  c _ж_ж1?) F A; g? g0   @`z  c _ж_ж1?F ) ; 2m+1=prime (Constantine)^gg ggff3   @`  c ` _ж_ж1?F ; F K2m+1 (spanning unicyclic)eg$o$o$o$g$ g gg&     @`#   c _ж_ж1?AYF  g? g0   @`   c `,_ж_ж1?) AF   3 (Krussel, et.al.)rggggff3gff3@       @`   c F_ж_ж1?) F  8m=2k or 5"2k (Constantine)ggogogff3   @`   c XT_ж_ж1?F   K2m (spanning tree)g g$o$o$g$g   @`G   c M_ж_ж1?AY  Given (Isomorphic)2e g   @`:  c c_ж_ж1?) A ~Given 2eg   @`J  c k_ж_ж1?)  Assigned (Isomorphic)2 eg   @`8  c w_ж_ж1? | coloring  g   @`B  To ?Y~B  N1 ?F YF B  To ?;Y;B  To ?;~B  N1 ?) ) ;~B  N1 ?AA;B  To ?YY;B  To ?;B  To ?YB  To ?H  0޽h ? C7"IDqIi+z2O___PPT10i. ]+D=' m= @B +6 ! 0 ME(    < ,YS     Z19=M n Introduction$ 2 g   5 Y; # #"]Y; #  c "_ж_ж1?AF Y; g2 g   @`  c t_ж_ж1?) F A; ^ g0   @`  c $t_ж_ж1?F ) ; ^ gff3   @`  c t_ж_ж1?F ; F K2m+1 (spanning unicyclic)eg$o$o$o$g$ g gg&     @`   c t_ж_ж1?AYF  3 trees if m>13rggggg   @`<   c h:_ж_ж1?) AF   6ggff3   @`   c в:_ж_ж1?) F  `"gff3   @`   c _ж_ж1?F   K2m (spanning tree)g g$o$o$g$g   @`G   c G,_ж_ж1?AY  Given (Isomorphic)2e g   @`:  c :_ж_ж1?) A ~Given 2eg   @`J  c N(_ж_ж1?)  Assigned (Isomorphic)2 eg   @`8  c 81_ж_ж1? | coloring  g   @`B  To ?Y~B  N1 ?F YF B  To ?;Y;B  To ?;~B  N1 ?) ) ;~B  N1 ?AA;B  To ?YY;B  To ?;B  To ?YB  To ?  ZDt1p w  T  2f0   Zz:1B  I T  2f0 ~  BA ? ? x  ~  BA ? ?h +  H  0޽h ? C7"IDqIi+z2O___PPT10i. ]+D=' m= @B + ! 0 $0(  $x $ c $,YS   x $ c $.Y   H $ 0޽h ? C7"IDqIi+z2O___PPT10i. ~+D=' m= @B +  ! 0 kc ((  ( ( S /Y<$@ 0   ( 6( , S   ' ( Z811kM !Multicolored Subgraph Parallelism$" 2"g4       ( BA ? ? 8 $@ 0 ( BA ? ?  8 $D 0H ( 0޽h ? C7"IDqIi+z2O:2___PPT10.08+8mD' m= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*(%(+ ! 0  ,(  , , <F YP   , 0\H H@8___PPT9 OcProve the case that m is an odd integer. By Theorem 2.1 and Lemma 2.2 , we can conclude the proof.d% 2ggg gg gg d 8X  , 6M , S   '  , ZN1kM !Multicolored Subgraph Parallelism$" 2"g4      H , 0޽h ? C7"IDqIi+z2O___PPT10i.`cr+D=' m= @B +} ! 0  $(   r   S b,YS   r   S cY   H   0޽h ? C7"IDqIi+z2O___PPT10i. ~+D=' m= @B + ! 0 (    S iY<$D 0    6Pj , S   '  Zk1kM !Multicolored Subgraph Parallelism$" 2"g4      H  0޽h ? C7"IDqIi+z2O___PPT10n.@ o+YDB' m= @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(+ ! 0 k(  r  S wY    6 , , S   '  Zh1kM !Multicolored Subgraph Parallelism$" 2"g4      H  0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' m= @B +A ! 0 XP(     <, ,YS      B\, , ,    0 H@8___PPT9 %By induction on v. Using Lemma 2.4. |&% 2gg g gg &  @`'   Z 1kM !Multicolored Subgraph Parallelism$" 2"g4       0  `A ? ?@{    H  0޽h ? C7"IDqIi+z2O___PPT10i.5kp+D=' m= @B +t ! 0 #'08(  0 0 S lS <$D  0   0 6 , S   ' 0 Z1kM !Multicolored Subgraph Parallelism$" 2"g4      l , '0,,$D 0TB  0B c $DjJFS  t TB  0 c $DjJFt V wTB  0 c $DjJV w wTB  0 c $DjJ t wTB  0 c $DjJ S t TB 0B c $DjJV S  wTB 0 c $DjJ S  wTB 0 c $DjJFt  wTB 0 c $DjJFt t TB 0 c $DjJV t w 0 0 Xw , S1" 2g   0 0l w  S1" 2g   0 0 l  S2" 2g   0 0c a  S2" 2g   0 0 p  S3" 2g   0 0,  S3" 2g   0 0I 8  S4" 2g   0 0d .  S4" 2g   0 08 W  S5" 2g   0 0 ~ R  S5" 2g  2 0 Z1"` w T 2 0 Z1"`[r 2 0 Z1"` d 2  0 Z1"` G 2 !0 Z1"`   "0 0  o  S1" 2g   #0 0( F  S2" 2g   $0 0Xr , S3" 2g   %0 0" X , S4" 2g   &0 0&   S5" 2g  H 0 0޽h ? C7"IDqIi+z2Ot l ___PPT10L .0%u+䷦D ' m= @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*06%(D' =-s6Bwipe(down)*<3<*06D' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*06Y%(D' =-s6Bwipe(down)*<3<*06YD4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'0%(+c~ ! 0 CC_zPC(  P P 6L7 , S   ' P Z21kM !Multicolored Subgraph Parallelism$" 2"g4       P 04 :  8   kP  P 0;Y S1" 2g  2 P Z1"`Y2 P Z1"``Y  P 0?`X4 S2" 2g  2  P Z1"` Y   P 0xC Y  S3" 2g  2  P Z1"`Y Z   P 0pGV Y*  S4" 2g  2 P Z1"`Y P 0KQ S5" 2g  B P TD1`B P TD1 B P TD1 V B P TD1  P c BCDEFAA1[|@  Y@6  P s BCDEF1[|@  Y6 , P s BCDEF1[|@  Y  P c BCqDEFAA1[={q@  I@   P c BkC DEFAA1kODk @  @, !P s BCqDEF1[={q@  I, "P 0RP$ S4" 2g   #P 0 V`4 S2" 2g   $P 0YV *  S4" 2g   %P 0x]   S3" 2g   &P 0`    S3" 2g   'P 0d v S2" 2g   (P 0Lc'   S1" 2g   )P 0l  S5" 2g   *P 0pQw%  S5" 2g   +P 0\t w  S1" 2g  Bl  m lP m,$@ 02 ,P T1"`  m2 >P T1"` Dm2 HP T1"`FmBl   mP ,$@ 02 8P T1"`  2 ?P T1"` D2 IP T1"`FBl   c  nP c ,$@ 02 :P T1"`  c 2 @P T1"`  Dc 2 JP T1"`F c Bl    oP  ,$@ 02 +B#style.visibility<*lP%(D' =+4 8?bCB#ppt_x-.2BCB#ppt_xB*Y3>B ppt_x<*lPD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*lPD' =-6Bwipe(right)*q3=-B%gradientSize: 0.1<*lPD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*mP%(D' =+4 8?bCB#ppt_x-.2BCB#ppt_xB*Y3>B ppt_x<*mPD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*mPD' =-6Bwipe(right)*q3=-B%gradientSize: 0.1<*mPD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*nP%(D' =+4 8?bCB#ppt_x-.2BCB#ppt_xB*Y3>B ppt_x<*nPD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*nPD' =-6Bwipe(right)*q3=-B%gradientSize: 0.1<*nPD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*oP%(D' =+4 8?bCB#ppt_x-.2BCB#ppt_xB*Y3>B ppt_x<*oPD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*oPD' =-6Bwipe(right)*q3=-B%gradientSize: 0.1<*oPD' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*pP%(D' =+4 8?bCB#ppt_x-.2BCB#ppt_xB*Y3>B ppt_x<*pPD' =+4 8?\CB#ppt_yBCB#ppt_yB*Y3>B ppt_y<*pPD' =-6Bwipe(right)*q3=-B%gradientSize: 0.1<*pPD' =%(DM' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*vP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*wP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*xP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*yP%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*zP%(D' =%(D{' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*qP%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*qPD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*qPD' =-g6B fade*<3<*qPDg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*rP%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*rPD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*rPD' =-g6B fade*<3<*rPDg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*sP%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*sPD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*sPD' =-g6B fade*<3<*sPDg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*tP%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*tPD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*tPD' =-g6B fade*<3<*tPDg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*uP%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*uPD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*uPD' =-g6B fade*<3<*uP++0+vP0 ++0+wP0 ++0+xP0 ++0+yP0 ++0+zP0 +j ! 0 WWy)mW(    < ,YS     6t    '  Z1kM !Multicolored Subgraph Parallelism$" 2"g4      2 # T1"` m2 $ T1"` 2 % T1"` c 2 & T1"` 2 ' T1"` Y2 ( T1"` Dm2 ) T1"` D2 * T1"` Dc 2 + T1"` D 2 , T1"` DY2 - T1"`Fm2 . T1"`F2 / T1"` Fc 2 0 T1"` F 2 1 T1"`FY`l  3tS  !m t,$D 0B  TD1 3 S B  TD13S B  TD1t3tS B  TD1 3S B  TD13tS B  TD1 3tS B  TD1 3tS B  TD1 3S B  TD13tS Xl  JJ " JJ,$D 0 J 0vJ T10" 2g   K 0|  n T11" 2g   L 0:1J T12" 2g    0 "  T10" 2g    0 T10" 2g    0\CRS& T11" 2g    0 ^ 2 T11" 2g    0& #  T12" 2g    0$* ^# 2 T12" 2g  F    D  0,Y S1" 2g  2  Z1"`Y2  Z1"``Y  0\2`X4 S2" 2g  2  Z1"` Y   06 Y  S3" 2g  2  Z1"`Y Z   0;V Y*  S4" 2g  2  Z1"`Y  0?Q S5" 2g  B  TD1`B  TD1 B  TD1 V B  TD1   s BCDEF1[|@  Y@6   s BCDEF1[|@  Y6 ,  s BCDEF1[|@  Y   s BCqDEF1[={q@  I@   s BkC DEF1kODk @  @,  s BCqDEF1[={q@  I,  0GP$ S4" 2g    0K`4 S2" 2g    0pJV *  S4" 2g    0T   S3" 2g    0Y    S3" 2g    0|\ v S2" 2g    0Y'   S1" 2g    0c  S5" 2g    0hQw%  S5" 2g    0l w  S1" 2g  l     `,$D 0B  TD8c    0p   S4" 2g   := ޔ# #":=,$D 0& ߔ c Dw_ж_ж1 ?<= j12"g   @`&  c h_ж_ж1 ?;< j10"g   @`&  c _ж_ж1 ?:; j11"g   @`&  c x_ж_ж1 ?<= j10"g   @`&  c _ж_ж1 ?;< j11"g   @`&  c _ж_ж1 ?:; j12"g   @`&  c <_ж_ж1 ?<= j11"g   @`&  c _ж_ж1 ?;< j12"g   @`&  c _ж_ж1 ?:; j10"g   @`B  Z1 ?:=B  T1 ?:=B  T1 ?:=B  Z1 ?:=B  Z1 ?::B  T1 ?;;B  T1 ?<<B  Z1 ?== :p=  ) #":=,$D 0  c ,_ж_ж1 ?< =  b$g   @`  c _ж_ж1 ?; <  b$g   @`(  c _ж_ж1 ?: ;  l11$g   @`(  c _ж_ж1 ?< =  l10$g   @`  c  _ж_ж1 ?; <  b$g   @`  c _ж_ж1 ?: ;  b$g   @`  c _ж_ж1 ?<p=  b$g   @`(  c  _ж_ж1 ?;p<  l12$g   @`  c d_ж_ж1 ?:p;  b$g   @`B  Z1 ?:p;pB  Z1 ?: ; B  Z1 ?:p: B  Z1 ?=p= B  Z1 ?;p<pB  Z1 ?: : B  Z1 ?<p=pB   Z1 ?= = B   Z1 ?: : B  Z1 ?= = B  Z1 ?; < B  Z1 ?< = dl  c t  $m t,$D 0  0t*l @  T10" 2g  B  TD8c c  B  TD8cc t B  TD8c c t   0. T (  T11" 2g     0x4 #  T12" 2g  B # HDjJ} },$D 0H  0޽h ? C7"IDqIi+z2Otl___PPT10L.0%u+ ~D ' = @B D' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*!%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*ޔ%(D' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*#%(D' =-6B+checkerboard(across)*<3<*#D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*"%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*)%(D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*$%(D' =-o6Bdissolve*<3<*$+\ ! 0 /O'Ov N(    6P , S   '  Zt1kM !Multicolored Subgraph Parallelism$" 2"g4        0L :  2 # T1"`  m2 $ T1"`  2 % T1"`  c 2 & T1"`  2 ' T1"`  Y2 ( T1"` dm2 ) T1"` d2 * T1"` dc 2 + T1"` d 2 , T1"` dY2 - T1"`fm2 . T1"`f2 / T1"` fc 2 0 T1"` f 2 1 T1"`fYF D  Z2 R  `1"`2 S  `1"``2 U  `1"`  2 W  `1"`Y  2 Y  `1"`B [ ZD177`B \ ZD177 B ] ZD17 7V B ^ ZD17 7 _  BCDEF1[|@  @G6  `  BCDEF1[|@  6 G, a  BCDEF1[|@  G  b  BCqDEF1[={q@  @  c  BkC DEF1kODk @  @, d  BCqDEF1[={q@  , e 6i P$ S4" 2g   f 6e`4 S2" 2g   g 6\lV *  S4" 2g   h 6t   S3" 2g   i 6xD  S3" 2g   j 6|  v S2" 2g   k 6< '   S1" 2g   l 6   S5" 2g   m 6Q%  S5" 2g   n 6(   S1" 2g   z g<=(  <Z(,$D 0 {  BCDEF8c[|@  "`<2  |  BCDEF8c[|@  "`2 ( }  BCDEF8c[|@  "`  ~  BCqDEF8c[={q@  "`<J    BCqDEF8c[={q@  "`J(  6\=0 S2" 2g    6R =&  S4" 2g    6 =  S3" 2g    6gM!  S5" 2g    6dg   S1" 2g   l  J,$D 0B  ZD8c`B  ZD8c B  ZD8c V B  ZD8c    BkC DEF8ckODk @  "`@,  6P$ S4" 2g    6Ȭ   S3" 2g    6t v S2" 2g    6l'   S1" 2g     60  S5" 2g  2  Z1"`k2  Z1"`_k2  Z1"` k 2  Z1"`V k 2  Z1"`kl  mt m ,$D 0B  TD1 m B  TD1mB  TD1tmtB  TD1 mB œ TD1mtB Ü TD1 mB Ĝ TD1mtB Ŝ TD1 mtB Ɯ TD1 mtB ǜ TD1   B Ȝ TD1 B ɜ TD1tt B ʜ TD1  B ˜ TD1t B ̜ TD1  B ͜ TD1t B Μ TD1 t B Ϝ TD1 t B М TD1 c B ќ TD1c  B Ҝ TD1tc t B Ӝ TD1 c  B Ԝ TD1c t B ՜ TD1 c  B ֜ TD1c t B ל TD1 c t B ؜ TD1 c t B ٜ TD1 B ڜ TD1 B ۜ TD1t tB ܜ TD1 B ݜ TD1 tB ޜ TD1 B ߜ TD1 tB  TD1 tB  TD1 tB  TD1 mB  TD1mtB  TD1 mB  TD1mt  c BC DEFAA1^"o @   mt  c BC DEFAA1X1a @   mt  c BC DEFAA1q1aq @   m   s BC DEF1q1aq @  3 m  s BC DEF1q1aq @  mtH  0޽h ? C7"IDqIi+z2O  ___PPT10m .0%u+:SDA ' = @B D ' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Dd' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*+> ! 0 44Nc4(    6 , S   '  ZT1kM !Multicolored Subgraph Parallelism$" 2"g4        0T :  2  T1"`  m2  T1"`  2  T1"`  c 2  T1"`  2   T1"`  Y2   T1"` dm2   T1"` d2   T1"` dc 2   T1"` d 2  T1"` dY2  T1"`fm2  T1"`f2  T1"` fc 2  T1"` f 2  T1"`fY F  9 qB : TD8c`B ; TD8c B < TD8c V B = TD8c  >  BkC DEF8ckODk @  "`@, ? 0P$ S4" 2g   @ 0   S3" 2g   A 0 v S2" 2g   B 0X'   S1" 2g   C 0x  S5" 2g  2 D Z1"`k2 E Z1"`_k2 F Z1"` k 2 G Z1"`V k 2 H Z1"`kbl t1 i 1t i,$D 0 v ZD1t1  jBipartite difference = 0" 2g  B w ZD8c=   x Z1t S  jBipartite difference = 0" 2g  B y ZD8c=   z Z1t  jBipartite difference = 0" 2g  B { ZD8c= V V  | Z 1t v jBipartite difference = 0" 2g  B } ZD8c=  ~ Z1t i jBipartite difference = 2" 2g  B  ZD8c=  (  Z1t,$ 0 jC(1)6 2em  )l  mw m w,$D 0B I ZD1 m B J ZD16m6B K ZD1mB R ZD1   B S ZD166 B T ZD1 B [ ZD1 c B \ ZD16c 6 B ] ZD1c  B d ZD1 B e ZD16 6B f ZD1   6 "  T10" 2g    64 " T11" 2g    6(#g"w T12" 2g    6@'P  v S4" 2g    6* dv S5" 2g    6g v S6" 2g    60P    S1" 2g    6t4 d  S2" 2g    64:g   S3" 2g    6C   T13" 2g    6H   T14" 2g    6Rgw  T15" 2g    6V   S9" 2g    6lZE   S7" 2g    6^ t  S8" 2g  B  TD1 m6B  TD16m  s BC DEF1X1a @   mH  0޽h ? C7"IDqIi+z2O  ___PPT10 .0%u+(D9 ' = @B D' = @BA?%,( < +O%,( < +D' =%(D' =%(DG' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B'blinds(horizontal)*<3<*D' =%(D' =%(D3' =4@BB BB%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-o6Bdissolve*<3<*DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(+8+0+0 +2 ! 0 11 LqU1(    6n , S   '  Zo1kM !Multicolored Subgraph Parallelism$" 2"g4        0ps :  2  T1"`  m2  T1"`  2  T1"`  c 2  T1"`  2   T1"`  Y2   T1"` dm2   T1"` d2   T1"` dc 2   T1"` d 2  T1"` dY2  T1"`fm2  T1"`f2  T1"` fc 2  T1"` f 2  T1"`fY F   qB  TD8c`B  TD8c B  TD8c V B  TD8c    BkC DEF8ckODk @  "`@,  0}P$ S4" 2g    0~   S3" 2g    0 v S2" 2g    0Ȃ'   S1" 2g    0P  S5" 2g  2  Z1"`k2   Z1"`_k2 ! Z1"` k 2 " Z1"`V k 2 # Z1"`k|B ' TD1m 6|B ( TD1m6|B , TD1m |B 0 TD1 6 |B 1 TD16 |B 5 TD1  |B 9 TD1c 6 |B : TD1c 6 |B > TD1c  |B B TD1 6|B C TD1 6|B G TD1  Q Z11t  jBipartite difference = 1" 2g  B R ZD8c=  S Z1t S  jBipartite difference = 1" 2g  B T ZD8c=  U Z1 t  jBipartite difference = 1" 2g  B V ZD8cV = V  W Z1 t v jBipartite difference = 1" 2g  B X ZD8c =  Y Zܠ1t i jBipartite difference = 1" 2g  B Z ZD8c= |B \ TD1m 6|B ] TD1m6 ^ s BC DEF1^"o @  m  _ 0ԥ 2 s  S3" 2g   ` 0' (  S1" 2g   a 0 S   S2" 2g   b 0LH  S6" 2g   c 0 W S4" 2g   d 0u T I S5" 2g   e 08  T12" 2g   f 0ܿt` T10" 2g   g 0#7F T11" 2g   k 0$p` pD  T15" 2g   l 04   T13" 2g   m 0l  T14" 2g   n 0 C  S9" 2g   o 0 s)  S7" 2g   p 0X S   S8" 2g   q Z1t jC(2)6 2em  H  0޽h ? C7"IDqIi+z2O___PPT10i.0%u+D=' = @B +3 ! 0 220Lm-2(    6 , S   '  Z1kM !Multicolored Subgraph Parallelism$" 2"g4        0 :  2  T1"`  m2  T1"`  2  T1"`  c 2  T1"`  2   T1"`  Y2   T1"` dm2   T1"` d2   T1"` dc 2   T1"` d 2  T1"` dY2  T1"`fm2  T1"`f2  T1"` fc 2  T1"` f 2  T1"`fY F   qB  TD8c`B  TD8c B  TD8c V B  TD8c    BkC DEF8ckODk @  "`@,  0P$ S4" 2g    0D   S3" 2g    0 v S2" 2g    0'   S1" 2g    0x   S5" 2g  2  Z1"`k2   Z1"`_k2 ! Z1"` k 2 " Z1"`V k 2 # Z1"`k|B ) TD1m 6|B * TD1m6|B + TD1m |B 2 TD1 6 |B 3 TD16 |B 4 TD1  |B ; TD1c 6 |B < TD1c 6 |B = TD1c  |B D TD1 6|B E TD1 6|B F TD1  N s BC DEF1q1aq @  m  O s BC DEF1q1aq @  mS6 P s BC DEF1q1aq @  m Q Z11t  jBipartite difference = 2" 2g  B R ZD8c=  S ZT1t S  jBipartite difference = 2" 2g  B T ZD8c=  U Z1 t  jBipartite difference = 2" 2g  B V ZD8cV = V  W Z@1 t v jBipartite difference = 2" 2g  B X ZD8c =  Y Z "1t i jBipartite difference = 0" 2g  B Z ZD8c=  [ 0p' pg S6" 2g   \ 0*u 6I S5" 2g   ] 0<.u I S4" 2g   ^ 02 q  S3" 2g   _ 0l6 7  S2" 2g   ` 0: W  S1" 2g   a 0= m #  S7" 2g   b 0XA   S8" 2g   c 0D )  S9" 2g   g 0H%` q T12" 2g   h 0DL7F T11" 2g   i 0O  T10" 2g   j 0S` q  T15" 2g   k 00W7Fb  T14" 2g   l 0Z$ T13" 2g   m Zx^1t jC(3)6 2em  H  0޽h ? C7"IDqIi+z2O___PPT10i.0%u+D=' = @B +y  ! 0 E=$t(  tx t c $l   t 6m , S   ' t Zn1kM !Multicolored Subgraph Parallelism$" 2"g4      \ $t Ztt1t, ,$D 0 K5(3) has been partitioned into 3 multicolored Hamiltonian cycles. Remaining C3 s ?V( 2emee#e g g o g  V H t 0޽h ? C7"IDqIi+z2O___PPT10.0%u+j܏D' = @B DC' = @BA?%,( < +O%,( < +Dz' =%(D"' =%(D' =4@BB%BB%(D' =1:Bvisible*o3>+B#style.visibility<*$tEV%(D' =-g6B fade*<3<*$tEVD' =+4 8?\CB#ppt_xBCB#ppt_xB*Y3>B ppt_x<*$tEVD-' =+4 8?hCB#ppt_y+1BCB#ppt_y-.03B*Y3>B ppt_y<*$tEV)?DI' d=+4 8?dCB#ppt_y-.03BCB#ppt_yB*Y3>B ppt_y<*$tEV%()?+[ ! 0 ;;XXx;(    6 , S   '  Z<1kM !Multicolored Subgraph Parallelism$" 2"g4        0<   2  T1"`[m2  T1"`[2  T1"` [c 2  T1"` [ 2   T1"`[Y2   T1"`^ m2   T1"`^ 2   T1"` ^ c 2   T1"` ^ 2  T1"`^ Y2  T1"` m2  T1"` 2  T1"` c 2  T1"`  2  T1"` YaF  mw  mJB  TD1 m B  TD16m6B  TD1mB  TD1   B  TD166 B  TD1 B  TD1 c B  TD16c 6 B  TD1c  B  TD1 B  TD16 6B   TD1  ! 0 "  T10" 2g   " 0 " T11" 2g   # 0g"w T12" 2g   $ 0P  v S4" 2g   % 0| dv S5" 2g   & 0,g v S6" 2g   ' 0ܱP    S1" 2g   ( 0 d  S2" 2g   ) 0g   S3" 2g   * 0   T13" 2g   + 0̿   T14" 2g   , 0pgw  T15" 2g   - 0   S9" 2g   . 0E   S7" 2g   / 0\ t  S8" 2g  B 0 TD1 m6B 1 TD16m 2 s BC DEF1X1a @   m 3 Z1p   |C(1)H 2eme  z  /t 4 /, ,$@ 0B 5 TD1 @ @B 6 TD1C@G@ 7 s BC[DEF1[-?^}-[@   t 8 0   S1" 2g   9 0S  S2" 2g   : 0 /p S3" 2g  z  ta ; , a,$@ 0B < TD1  B = TD1CG > s BC[DEF1[-?^}-[@   2t ? 0  a S4" 2g   @ 0dS a S5" 2g   A 0| p~ S6" 2g  z  % t  B % , ,$@ 0B C TD1 6 6 B D TD1C6 G6  E s BC[DEF1[-?^}-[@   t  F 0   S7" 2g   G 0tS   S8" 2g   H 0 % p  S9" 2g  z  tW  I  , W ,$@ 0B J TD1 B K TD1C G  L s BC[DEF1[-?^}-[@   ( t  M 0d  W  T10" 2g   N 0S cW  T11" 2g   O 0 h  T12" 2g  z  t P , ,$@ 0B Q TD1 , ,B R TD1C,G, S s BC[DEF1[-?^}-[@   t T 0   T13" 2g   U 0 Sc T14" 2g   V 0  T15" 2g  X W Z<1S s ,$ 0 +5C3X 2eemi   X Z1 ': ,$ 0 [ = 4-factor. 2 e  H  0޽h ? C7"IDqIi+z2Ojb___PPT10B.0%u+D' "= @B Da' = @BA?%,( < +O%,( < +DO' =%(D' =%(Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*4%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*4D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*4D' =-g6B fade*<3<*4Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*;%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*;D' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*;D' =-g6B fade*<3<*;Dg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*BD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*BD' =-g6B fade*<3<*BDg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*I%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*ID' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*ID' =-g6B fade*<3<*IDg' =4@BB7BB%(D' =1:Bvisible*o3>+B#style.visibility<*P%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*PD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*PD' =-g6B fade*<3<*PDt' =A@BB7BB0B%(D' =1:Bvisible*o3>+B#style.visibility<*W%(D' =+4 8?fCB#ppt_w*0.70BCB#ppt_wB*Y3>B ppt_w<*WD' =+4 8?\CB#ppt_hBCB#ppt_hB*Y3>B ppt_h<*WD' =-g6B fade*<3<*WDA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*X%(+p+0+W0 ++0+X0 +DZ ! 0 BB@c0 B(    60 , S   '  Z 31kM !Multicolored Subgraph Parallelism$" 2"g4        05   2 4 T1"`8/2 6 T1"` 8% 2 7 T1"`E 8 2 8 T1"`82 9 T1"`;/2 : T1"`O;2 ; T1"` ;% 2 < T1"`E ; 2 = T1"`;2 > T1"`/2 ? T1"`O2 @ T1"` % 2 A T1"`E  2 B T1"`B F ZD1/OB H ZD1hh B L ZD1% E B N ZD1 hh R 6?  T12" 2g   T 6Bd8 S5" 2g   X 6E O  S3" 2g   Z 6Jn ;KB  T14" 2g   ] 6$N w-q  S7" 2g  B ` ZD1/hB c ZD16: e  BC[DEF1[-?^}-[@  y  f 6QF S1" 2g   h 6U  S3" 2g  B j ZD1}6:} m 6YOF#  S4" 2g   o 6]T ( S6" 2g  B q ZD1 6:  s  BC[DEF1[-?^}-[@  o   t 6Xa F  S7" 2g   v 6e :  S9" 2g  B x ZD1s 6:s  z  BC[DEF1[-?^}-[@   E  { 6@iE FW T10" 2g   } 6lV *  T12" 2g  B  ZD16:   BC[DEF1[-?^}-[@  e   6(qFW T13" 2g    6t T15" 2g  2  T1"` C/2  T1"`O C2  T1"` C% 2  T1"`E C 2  T1"` C2  T1"`F/2  T1"`OF2  T1"` F% 2  T1"`E F 2  T1"`F2  T1"`/2  T1"`O2  T1"` % 2  T1"`E  2  T1"`B  ZD1 /OB  ZD1 /ssOB  ZD1  B  ZD1  B  ZD1 % E B  ZD1 % ssE B  ZD1  B  ZD1    6$ 2 C T10" 2g    6 FV T11" 2g    6 d C8 S4" 2g    6 dZ8 S6" 2g    6@  C  S1" 2g    6    S2" 2g    6 n 2 CB  T13" 2g    6, n B  T15" 2g    6Н  $q  S9" 2g    6t b %6  S8" 2g  B  ZD1 /s   BC DEF1 X1a @  /B  ZD1  6e S2" 2g  B  ZD1}}  6Oe#  S5" 2g  B  ZD1     60  e  S8" 2g  B  ZD1 s s   6H E  T11" 2g  B  ZD1   6  T14" 2g  2  T1"`C7   BC[DEF1[-?^}-[@   CJ  Z41/z  O C(1)+5C3 2emeeme    0|FV,$ 0 V11$ 2g    0H ,$ 0 V11$ 2g    0h ,$ 0 V12$ 2g    0<W + ,$ 0 V12$ 2g  H  0޽h ? C7"IDqIi+z2O3+___PPT10 .0%u+nD' = @B Dj' = @BA?%,( < +O%,( < +Dh ' =%(D ' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*R%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*}%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D' =A@BBBB0B%(D' =1:Bhidden*o3>+B#style.visibility<*%(D1' =%(D' =%(D' =A@BBBB0B%()?)?D' =.7 BBBBBM 0 0 C 0.06597 0.01852 0.13194 0.03727 0.19583 0.03704 C 0.25972 0.0368 0.32152 0.01736 0.38333 -0.00185 *3>*B ppt_xB ppt_y=0BB aaA<*D' =A@BBBB0B%()?)?D/' =.7 BBBBBM -0.00226 -0.00232 C -0.06268 -0.01759 -0.12292 -0.03264 -0.18733 -0.03194 C -0.25157 -0.03125 -0.3198 -0.01505 -0.38802 0.00139 *3>*B ppt_xB ppt_y=B0BB aaApBB؂EBj[<*+0+0+R0 ++0+}0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 ++0+0 +R ! 0 ?F7F``E(    6p , S   '  Z1kM !Multicolored Subgraph Parallelism$" 2"g4        0   J _ Z1/z  O C(1)+5C3 2emeeme  2 d T1"` M 22 e T1"`R M 2 f T1"`P22 g T1"`R P 2 h T1"`22 i T1"`R  :F  3tS  j 2 R B k TD1 3 S B l TD13S B m TD1t3tS B n TD1 3S B o TD13tS B p TD1 3tS B q TD1 3tS B r TD1 3S B s TD13tS  u 64;o  T10" 2g   v 6_3 T11" 2g   w 6  T12" 2g   x 6Y j  T10" 2g   y 6Poo T10" 2g   z 6 T11" 2g   { 6#a q T11" 2g   | 6<z o T12" 2g   } 6!#z  T12" 2g  r := ~# #"WZ &  c %_ж_ж1 ?<= j12"g   @`&  c 5_ж_ж1 ?;< j10"g   @`&  c >_ж_ж1 ?:; j11"g   @`&  c G_ж_ж1 ?<= j10"g   @`&  c (C_ж_ж1 ?;< j11"g   @`&  c `Z_ж_ж1 ?:; j12"g   @`&  c c_ж_ж1 ?<= j11"g   @`&  c  l_ж_ж1 ?;< j12"g   @`&  c g_ж_ж1 ?:; j10"g   @`B  Z1 ?:=B  T1 ?:=B  T1 ?:=B  Z1 ?:=B  Z1 ?::B  T1 ?;;B  T1 ?<<B  Z1 ?==B  ZDjJBjz B W) ZY  #") WZY,$@ 0<  c Ԃ_ж_ж1 ?YIZY 118gg   @`&  c _ж_ж1 ?XIYY j10"g   @`&  c _ж_ж1 ?WIXY j12"g   @`&  c _ж_ж1 ?Y9 ZI j10"g   @`&  c T_ж_ж1 ?X9 YI j12"g   @`<  c _ж_ж1 ?W9 XI 118gg   @`&  c  _ж_ж1 ?Y) Z9  j12"g   @`&  c _ж_ж1 ?X) Y9  j11"g   @`&  c _ж_ж1 ?W) X9  j10"g   @`B  Z1 ?W) Z) B  T1 ?W9 Z9 B  T1 ?WIZIB  Z1 ?WYZYB  Z1 ?W) WYB  T1 ?X) XYB  T1 ?Y) YYB  Z1 ?Z) ZYl jV , V j,,$D 02  T1"` V M 2  T1"` M ,2  T1"`PV  2  T1"`P,2  T1"`V  2  T1"`,BN  3tS    B  TD1 3 S B  TD13S B  TD1t3tS B  TD1 3S B  TD13tS B  TD1 3tS B  TD1 3tS B  TD1 3S B  TD13tS   0Po  T10" 2g    0   T12" 2g    0Du I T11" 2g    0Y f j : T10" 2g    0o   T10" 2g    0 j T12" 2g    04a qv T12" 2g    0z   T11" 2g    0z v T11" 2g  B  ZDjJj z B   `DԔc s ,$@  0B   `DԔc pps ,$D  0H  0޽h ? C7"IDqIi+z2O  ___PPT10 .0%u+18D ' = @B D] ' = @BA?%,( < +O%,( < +D>' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#strips(downLeft)*<3<*DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B#strips(downLeft)*<3<*DN' =%(D' =%(DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*DK' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =-6B+checkerboard(across)*<3<*+j> ! 0 =y=P__=(    6e , S   '  Z1kM !Multicolored Subgraph Parallelism$" 2"g4        0|I   2  T1"`8/2  T1"` 8% 2  T1"`E 8 2  T1"`82   T1"`;/2   T1"`O;2   T1"` ;% 2   T1"`E ; 2   T1"`;2  T1"`/2  T1"`O2  T1"` % 2  T1"`E  2  T1"`|B  TD1/O|B  TD1hh |B  TD1% E |B  TD1 hh  0  V11$ 2g    0xd8 S5" 2g    0 O  U2$ 2g    0!n ;KB  T14" 2g    0$ w-q  U8$ 2g  |B  TD1/h|B  TD16:   BC[DEF1[-?^}-[@  y   0<(F S1" 2g     0\,  S3" 2g  |B ! TD1}6:} " 000OF#  S4" 2g   # 03T ( S6" 2g  |B $ TD1 6:  %  BC[DEF1[-?^}-[@  o   & 07 F  S7" 2g   ' 0; :  S9" 2g  |B ( TD1s 6:s  )  BC[DEF1[-?^}-[@   E  * 0?E FW T10" 2g   + 0CV *  T12" 2g  |B , TD16: -  BC[DEF1[-?^}-[@  e  . 0GFW T13" 2g   / 0pK T15" 2g  2 0 T1"` C/2 1 T1"`O C2 2 T1"` C% 2 3 T1"`E C 2 4 T1"` C2 5 T1"`F/2 6 T1"`OF2 7 T1"` F% 2 8 T1"`E F 2 9 T1"`F2 : T1"`/2 ; T1"`O2 < T1"` % 2 = T1"`E  2 > T1"`|B ? TD1/O|B @ TD1/ssO|B A TD1 |B B TD1 |B C TD1% E |B D TD1% ssE |B E TD1 |B F TD1  G 0V2 C T10" 2g   H 0ZFV V12$ 2g   I 0]d C8 S4" 2g   J 08bdZ8 S6" 2g   K 0e C  S1" 2g   L 0i   U3$ 2g   M 0$mn 2 CB  T13" 2g   N 0pn B  T15" 2g   O 0lt $q  S9" 2g   P 0xb %6  U7$ 2g  |B Q TD1/s R  BC DEF1X1a @  /|B S TD1 T 0{e S2" 2g  |B U TD1}} V 0Oe#  S5" 2g  |B W TD1   X 0̃ e  S8" 2g  |B Y TD1s s  Z 0E  T11" 2g  |B [ TD1 \ 0 T14" 2g  2 ] T1"`C7 ^  BC[DEF1[-?^}-[@   CJ _ ZЏ1/z  O C(1)+5C3 2emeeme  H  0޽h ? C7"IDqIi+z2O___PPT10i.0%u+D=' = @B + ! 0 p0(  x  c $x,YS   x  c $PY   H  0޽h ? C7"IDqIi+z2O___PPT10i. ~+D=' = @B + ! 0  r(    <ĩ ,YS   ~  s *l    Z1kM "Existence of Multicolored Subgraph$# 2#g(   H  0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B + ! 0 (    <̵ ,YS   ~  s *     Z1kM "Existence of Multicolored Subgraph$# 2#g(   2  T1"`P2  T1"`% 7 2  T1"` ) 2  T1"` t) 2   T1"`% W 2   T1"`% < 2   T1"` I2   T1"`% I 2   T1"`% ] |B @ TD1}d% |B @ TD1S d |B  TD1)  |B  TD1S d |B @ TD1}% |B  TD1}w% |B  TD1j3 |B  TD1} S   ZX1]:w} Ox 2a    Z1g ne1< 2aia    Z1}k  ne2< 2aia    Z1S ,s  ny1< 2aia    ZL1S  @ s  ny2< 2aia    Z1 W  nz1< 2aia    Z1S  s  nz2< 2aia  H  0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B +& ! 0 &&(    <, ,YS   ~  s *     ZP1kM "Existence of Multicolored Subgraph$# 2#g(   2  T1"`P2  T1"`% 7 2  T1"` ) 2  T1"` t) 2   T1"`% W 2   T1"`% < 2   T1"` I2   T1"`% I 2   T1"`% ] |B @ TD1}d% |B @ TD1S d |B  TD1)  |B  TD1S d |B @ TD1}% |B  TD1j3   Zt1]:w} Ox 2a    ZH1g ne1< 2aia    Z 1}k  ne2< 2aia    Z1S ,s  ny1< 2aia    Z1S  @ s  ny2< 2aia    Z1 W  nz1< 2aia    Z`1S  s  nz2< 2aia  p8 }  $} B  TD1}w% B  TD1} S B B TD1S w B  TD1jS S   Z\)1 ,$ 0 Fact: If (e1) =(y2z2) and (e2) =(y1z1), then T[x;y1,y2;z1,z2] is multicolored.,U 2eeeemeememeeeemeememe e eeeeeemeemeemeeme U l  5  %} ,$D 0B   TD8c_B ! TD8cB "B TD8c 5 B # TD8c, & Z|J1P   T[x;y1,y2;z1,z2]b 2gggggoggoggoggog  H  0޽h ? C7"IDqIi+z2O___PPT10y.`;Lp+שD' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(Df' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*$%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*%%(+8+0+0 +s ! 0 (    <_ ,YS   ~  s *H    Zp`1kM "Existence of Multicolored Subgraph$# 2#g(    0 TA ? ?   H  0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B + ! 0 Pr(    <q ,YS   ~  s *rV     Zt1kM "Existence of Multicolored Subgraph$# 2#g(   H  0޽h ? C7"IDqIi+z2O___PPT10i.`;Lp+D=' = @B + ! 0 tl$'(    << ,YS   ~  s *    ZD1kM "Existence of Multicolored Subgraph$# 2#g(   2  T1"` ks 2  T1"`M 2   T1"`pP 2   T1"` @ 6 2   T1"`S C 2   T1"` ] 9 2   T1"`  a F 2  T1"`6 { 2  T1"`s z 2  T1"` 3 9 2  T1"` = 2  T1"` & 2  T1"`) 2  T1"`) q   ZX14 q %  nx0< 2aia  ,  Z1 V  ,$ 0 nx1< 2aia  l   ' ,$D 0~B  ND1 % m ~B  ND1 @ ~B  ND1 m  ~B  ND1M  m c ~B  ND1= m F ~B B ND1  m ~B B ND1  m ~B   ND1m  V ~B ! ND1m  3  ~B " ND1m ~  ~B # ND1m  ~B $ ND1m   ~B % ND1m  CV ~B & ND1m   2  T1"` S H  0޽h ? C7"IDqIi+z2O___PPT10.`;Lp+FD4' = @B D' = @BA?%,( < +O%,( < +DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*%(D' =%(D' =%(D-' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*'%(D' =-i6B wedge*<3<*'+8+0+0 +9 ! 0 3t(    <p ,YS   ~  s *H m    Z1kM "Existence of Multicolored Subgraph$# 2#g(   f2 % 0"`0 d f2 ) 0"` t ,8    a  n2 ( 0"`  n2 * 0"`  f2 + 0"` d f2 , 0"` D f2 - 0"` $ f2 . 0"` d f2 / 0"`   6 0Ԭ@T:  nx0< 2go   7 0D@t :  x1R 2gog   8 0ض 4 ,$ 0 v1R 2gog  Cl [O I c [ ,$D 0 9 0dO I x2R 2gog   : 0[O KI ku: 2gg  l ` d  d` d ,$D  0ZB 3B s *D1`  ZB 4B s *D1t`   < 0x   U1$ 2g   = 0t d  U2$ 2g  f2 A 0"`;|f2 C 0"` k|f2 D 0"`[|f2 E 0"`|f2 F 0"`{|f2 G 0"`  |f2 H 0"` k |f2 N 0"`Yl  = h  =,$D   0B W TD1 = X 0d  U1$ 2g  l   i ,$@   0B VB TD1  Y 08" U2$ 2g  `l  F   b # ,$D 0n2 _ 0"` F i n2 ` 0"`)F  2 g 0"` E ,$@ 0l tY mYt,$@  0 \ 0t u1R 2gog  n2 j 0"`Yz C  t z ,,$D  0 Z 6C4___PPT10 B___PPT9$ v1'd 2gogc  t2 k 6"` Nc n Z1m` ,$ 0 Find a suitable v1.b 2ggog  f2 & 0"`0  H  0޽h ? C7"IDqIi+z2O___PPT10.`;Lp+KD' = @B D' = @BA?%,( < +O%,( < +D' =%(Df' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*a%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*b%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*c%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*n%(DW' =%(D' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*,%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*g%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*8%(D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*d%(D>' =%(D' =%(DC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*i%(D' =-6B#strips(downLeft)*<3<*iDC' =4@BBB B%(D' =1:Bvisible*o3>+B#style.visibility<*h%(D' =-6B#strips(downLeft)*<3<*hD' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*m%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*t%(+p+0+80 ++0+n0 +q4 ! 0 O#G#7F"(    < ,YS   ~  s *Tjm    Z1kM "Existence of Multicolored Subgraph$# 2#g(   f2  0"` t f2   0"` d f2   0"` $ f2   0"` d f2  0"`    0d C"?@T:  nx0< 2go    0dC"?@t :  x1R 2gog    0C"? 4  v1R 2gog    0C"?   x2R 2gog    0!C"? [K  ku: 2gg  f2  0"`;|f2  0"` k|f2  0"`[|f2  0"`|f2  0"`{|f2   0"` k |l  = B  =,$@  0B * TD1 = + 0P(C"?  U1$ 2g  l   A ,$D  0B )B TD1  , 0$,C"?" U2$ 2g   - 0l0C"?2{ ,4___PPT10 B___PPT9$ v1'P 2gof   . 04t u1R 2gog  B 1@ TD1S  ,$@  0l S @  CS @ ,$@  0B 0B TD1S S ZB B s *D1`  B /B TD1S  B 2B TD1ZS  B 3 TD1S z B 4 TD1S B 5 TD1S @ IB 7 TD1S IB 9B TD1S IB :B TD1 IB <B TD1wS IB =B TD1:S IB >B TD1* IB ?B TD1S f2   0"`  f2 " 0"`Yf2  0"`  |f2  0"`0   @ ZD1  e!Let T1(2)=T1(1)[x1;x2,v1;u1,v1']." 2eemmeemmeemeemeemeemeemge " f2  0"`  f2  0"`0 d l S   FS  ,$D  0B D TD1S  B EB TD1jS  f2   0"` D H  0޽h ? C7"IDqIi+z2O___PPT10.`;Lp+kDv' = @B D1' = @BA?%,( < +O%,( < +DS' =%(D' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C%(D' =-m6Bbox(in)*<3<*CD1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*1%(D' =-m6Bbox(in)*<3<*1D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =-m6Bbox(in)*<3<*FD' =%(D' =%(D/' =4@BBBB%(D' =-m6Bbox(in)*<3<*FD' =1:Bhidden*o3>+B#style.visibility<*F%(D&' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B%(D' =-s6Bwipe(down)*<3<*BD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*A%(D' =-s6Bwipe(down)*<3<*A+%& ! 0  0.I(    <c ,YS   ~  s *c"    Ze1kM "Existence of Multicolored Subgraph$# 2#g(   f2  0"`0 d f2  0"`0  f2  0"` t f2   0"` d f2   0"` D f2   0"` $ f2   0"` d f2  0"`    0m@T:  nx0< 2go    0|q@t :  x1R 2gog    0v 4  v1R 2gog    6{   x2R 2gog    04 [K  ku: 2gg  f2  0"`;|f2  0"` k|f2  0"`[|f2  0"`|f2  0"`{|f2  0"`  |f2   0"` k |y - 0h2{ ,4___PPT10 B___PPT9$ v1'P 2gof   . 0|t u1R 2gog  |B :@ TD1  ; 0C"?" U2$ 2g  l V *  HV * ,$D 0B 8B TD1   < 04V *  kc: 2gg  l2   6"`  *l  F,$@  0%@  CB > TD1Iw ? 0q kc: 2gg  n2 @ 0"`^| D 0J2,4___PPT10 B___PPT9$ u1'P 2gof  l V   GV  ,$D  0@ tV *  BtV *  + 0V *  U2$ 2g  B =B TD1  n2 A 0"`t ! h E 0  4___PPT10 B___PPT9$ ju'8 2gf  f2  0"`  f2 " 0"`YH  0޽h ? C7"IDqIi+z2O___PPT10.`;Lp+GDp' = @B D+' = @BA?%,( < +O%,( < +D4' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H%(D&' =%(D' =%(D7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F%(D' =-s6Bwipe(down)*<3<*FD7' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G%(D' =-s6Bwipe(down)*<3<*G+H= ! 0  ((@<D'(    <\ ,YS     Z1kM "Existence of Multicolored Subgraph$# 2#g(   f2   0"` d f2   0"` $ f2   0"` d f2   0"`    0@T:  nx0< 2go    0@t :  x1R 2gog    0` 4  v1R 2gog    0 [K  ku: 2gg  f2  0"`;|f2  0"` k|f2  0"`|f2  0"`{|f2  0"` k |  02{ , v'P 2ggg     0Dt u1R 2gog   % 02J, u1'h 2gogg   & 0|   lu': 2gg  l S @ I 3S @ I,$@ 0B " TD1S Z B # TD1ZS  B $B TD1 Z B 'B TD1* ZIB (B TD1: ZIB )B TD1J Z B *B TD1w ZIB + TD1Z Z B , TD1Z IB - TD1Z j B . TD1Z z B / TD1Z IB 0 TD1Z IB 1 TD1Z B 2 TD1Z @ If2  0"`  |f2   0"` D f2  0"`0 d f2  0"`0  l I 9I,$@ 0B 5 TD1I 6 04:x Uc$ 2g  l    8  ,$D  0B 4 TD1   7 0   U2$ 2g  f2  0"`[|f2  0"` t f2  0"`  f2  0"`Yl  Z < Z,$D 0B : TD1 Z B ;B TD1 Z~ = Z1@  Let T2(2)= x2-star 2eemmeeme   > Zh1,$ 0 < x2u  x2u1 2eaemeeeememe   ? Z1% $ 08___PPT10N___PPT90( P+uu' +u1u1' . 2eegeeememff&    F       n2  0"`    0d   x2R 2gog  S C c $|  )Round 2. Use u, u1 to construct 2nd tree.*x eeeememe * H  0޽h ? C7"IDqIi+z2O___PPT10.`;Lp+AD'  = @B D' = @BA?%,( < +O%,( < +D' =%(D^' =%(D' =4@BBBB%()?)?DZ' =."7 BBBBB[M -4.44444E-6 2.96296E-6 L 0.11025 -0.12593 *3>*B ppt_xB ppt_y=0BBAA<*D' =%(D' =%(D-' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*3%(D' =-i6B wedge*<3<*3D-' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*<%(D' =-i6B wedge*<3<*<DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*>%(D' =%(D' =%(D/' =4@BBBB%(D' =-m6Bbox(in)*<3<*<D' =1:Bhidden*o3>+B#style.visibility<*<%(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*?%(D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*9%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*8%(+p+0+>0 ++0+?0 +# ! 0   P1 P (    <7 ,YS     Z91kM "Existence of Multicolored Subgraph$# 2#g(   f2  0"`0 d f2  0"`0  f2  0"` t f2  0"` d f2   0"` D f2   0"` $ f2   0"` d f2   0"`     0@@T:  nx0< 2go    0D@t :  x1R 2gog  f2  0"`;|f2  0"` k|f2  0"`[|f2  0"`|f2  0"`{|f2  0"`  |f2  0"` k |f2   0"`  f2 - 0"`  f2 . 0"`Yl  C#  0C # ,$@ 0  00M C# =  x2R 2gog  n2 / 0"` 6 s H  0޽h ? C7"IDqIi+z2O___PPT10.`;Lp+< Dr' = @B D-' = @BA?%,( < +O%,( < +D' =%(D`' =%(D' =4@BBBB%()?)?D\' =.$7 BBBBB]M -1.94444E-6 -3.33333E-6 L 0.28351 -0.20995 *3>*B ppt_xB ppt_y=0BBAA<* D' =%(Df' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*0%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<* %(D ' =%(D ' =%(D+' =4@BBBB%()?)?D' =.G7 BBBBBYM -1.11111E-6 3.33333E-6 L 0.16997 -0.3926 *3>*B ppt_xB ppt_y=@0BBAApBBݭ=BI<*D+' =4@BBBB%()?)?D' =.G7 BBBBBYM -1.11111E-6 3.33333E-6 L 0.19097 -0.3926 *3>*B ppt_xB ppt_y=@0BBAApBB9=BI<*-D+' =4@BBBB%()?)?D' =.G7 BBBBBYM -4.44444E-6 3.33333E-6 L 0.02987 -0.3507 *3>*B ppt_xB ppt_y=@0BBAApBBJt<B3<* D+' =4@BBBB%()?)?D' =.G7 BBBBBYM 3.33333E-6 -1.85185E-6 L 0.10659 -0.4625 *3>*B ppt_xB ppt_y=@0BBAApBBOZ=Bl<*.D-' =4@BBBB%()?)?D' =.I7 BBBBB[M 1.38889E-6 -2.22222E-6 L 0.05955 -0.43125 *3>*B ppt_xB ppt_y=@0BBAApBB33<B$\<*D-' =4@BBBB%()?)?D' =.I7 BBBBB[M 1.38889E-6 -2.22222E-6 L 0.01979 -0.47315 *3>*B ppt_xB ppt_y=@0BBAApBB"""<By@r<*+l ! 0 44`IK |4(      <] ,YS   ~   s *X^      Z_1kM "Existence of Multicolored Subgraph$# 2#g(   f2   0"` ef2   0"` Ef2   0"` %f2   0"` e   0lf9 U3  nx0< 2go     0dj9 u3  x1R 2gog  f2   0"`<uf2   0"`\uf2   0"`uf2   0"` l uf2   0"` f2   0"` uf2   0"`k1f2   0"`pf2   0"`5f2   0"`Mf2   0"`Xf2 !  0"` ,2 &   `1"`I <  *  Zp1m)m ,$ 0 Find suitable v1, v2. 2eemeeme   +  0y  ,$ 0 v1R 2gog   ,  00 4 ,$ 0 v2R 2gog  l S   B L  ,$@ 0B - B TD1jS   3  0XC"?;   U1$ 2g  l j @  C y kA ,$@ 0B . B TD1j @  4  0,;  U2$ 2g  l S  D L  ,$@ 0B /  TD1S  5  0X '  Q3  2c  l @ S \  E L A ] ,$@ 0B 0  TDxb>1@ S P  6  0H ' \  U4$ 2g   9  0@B<,$   0 u1R 2gog   :  0ěB.k<,$  0 u2R 2gog   ;  0ܠC"?BN^<$   04___PPT10 B___PPT9$ v1'P 2gof   <  0C"? 4 D $  04___PPT10 B___PPT9$ v2'P 2gof  l  I F B,$@  0B 7 B TD1 I =  0Ȧ0 I U1$ 2g  z    I  D Q a ,$@  0B 8  ZD1V V  >  6   U2$ 2g  l : jI G  ;kB,$@   0B 1  TD1: jI ?  0E U Q3  2c  l  jI H  kB,$@  0B 2 B TDxb>1 jI @  0u II U4$ 2g   J  Zp1~ ,$ 08___PPT10N___PPT90( d"Let : T1(3)=T1(2)[x1;x3,v1;u1,v1']# 2eemmeemmeemeemeemeemeemfe #  K  Z1p~  $ 08___PPT10N___PPT90( LT2(3)=T2(2)[x2;x3,v2;u2,v2'] 2emmeemmeemeemeemeemeemfe  f2   0"`) e f2   0"`)  F  C#    < $    0 C# =  x2R 2gog  n2   0"` 6 s nz  F   '   &,$D 0n2 (  0"` F i n2 )  0"`)F  f2   0"`  f2   0"`  t f2   0"` 4 f2   0"` luf2   0"`|uf2   0"`uQz [O I #   N,$@ 0 $  0O I x3R 2gog   %  0[O KI ku: 2gg  H   0޽h ? C7"IDqIi+z2O_7W7___PPT1077.`;Lp+FG9D5' m= @B D4' = @BA?%,( < +O%,( < +D' =%(Dh' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*# %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*' %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<** %(DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*B %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*D %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*+ %(D ' =%(DN ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*F %(D' =-m6Bbox(in)*<3<*F D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*G %(D' =-m6Bbox(in)*<3<*G D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*; %(D' =-m6Bbox(in)*<3<*; D>' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*9 %(D' =-m6Bbox(in)*<3<*9 D' =%(Dd' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*B %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*D %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*J %(D' =%(Dd' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*F %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*G %(DY' =%(D' =%(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*E %(D' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*C %(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*, %(D ' =%(DN ' =%(D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*H %(D' =-m6Bbox(in)*<3<*H D1' =4@BBBB%(D' =1:Bvisible*o3>+B#style.visibility<*I %(D' =-m6Bbox(in)*<3<*I 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' =%(Dd' =%(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*E %(D' =4@BBBB%(D' =1:Bhidden*o3>+B#style.visibility<*C %(DA' =%(D' =%(D' =A@BBBB0B%(D' =1:Bvisible*o3>+B#style.visibility<*K %(++0+* 0 ++0++ 0 ++0+, 0 ++0+9 0 ++0+: 0 ++0+; 0 ++0+< 0 ++0+J 0 ++0+K 0 +c ! 0 >=Pb=(    <$ ,YS   ~  s *$     Z$&1kM "Existence of Multicolored Subgraph$# 2#g(   f2  0"`) e f2  0"`)  f2  0"`  t f2   0"` Ef2   0"` %f2   0"` ef2   0"`     0-9 U3  nx0< 2go    0829 u3  x1R 2gog  f2  0"`<uf2  0"`\uf2  0"`uf2  0"`uf2  0"` f2  0"` 4 f2  0"` uf2  0"`k1f2  0"`pf2  0"`5f2  0"`Mf2   0"`Xf2 ! 0"` ,F [O I "  N # 0<O I x3R 2gog   $ 08A[O KI ku: 2gg  2 %  `1"`I < c ) ZxE1m)m  Find suitable v1, v2. 2eemeeme   * 0N   v1R 2gog   + 0LS 4  v2R 2gog  z S   , L  ,$@  0B -B TD1jS   . 0(YC"?;   U1$ 2g  z j @  / y kA ,$D  0B 0B TD1j @  1 0,^;  U2$ 2g   8 0bB< u1R 2gog   9 0@fB.k< u2R 2gog  F : jI B  ;kBB C TD1: jI D 0kE U Q3  2c  F  jI E  kBB FB TDxb>1 jI G 0ou II U4$ 2g   H Zv1 I8___PPT10N___PPT90( d"Let : T1(3)=T1(2)[x1;x3,v1;u1,v1']# 2eemmeemmeemeemeemeemeemfe # | I Z<1p  8___PPT10N___PPT90( LT2(3)=T2(2)[x2;x3,v2;u2,v2'] 2emmeemmeemeemeemeemeemfe  8 ^H n LH ^n~B J ND1^ n  K 0ȟH  Uc$ 2g  F  C#   < $   08 C# =  x2R 2gog  n2  0"` 6 s Al MM e UMM e,$@  0 Q  BCDEFAA8cD/D@  "`MM  R 0e Q3  2c  l WZ g. VZ Wg.,$@   0B S TDxb>8cW g  T 0@zZ :. U4$ 2gxb>   W 0PC"?B0 <$  04___PPT10 B___PPT9$ u2'P 2gof   X 0dC"?Y'eS$   04___PPT10 B___PPT9$ u1'P 2gof   Y 0C"?9 G3$   04___PPT10 B___PPT9$ ju'8 2gf  l Cz \Cz,$@  0B Z TD8cECE [ 06z Uc$ 2g   ] ZT1  ,$ 0 T3(3)= x3-star 2emmeeeme  ^ ^ Z1 ` ,$ 0 $ x3u  x3u1 x3u12 2eaemeeememdemem&    _ Zh1 Z( $ 08___PPT10N___PPT90( +uu' +u1u1' +u2u2'` 2eegeeememfddememd&   l H  bH :,$D 0B ` TD1   a 0H  kc: 2gg  f2  0"` luf2  0"`|uf2  0"` l uf2  0"` e:F  F   &  %n2 ' 0"` F i n2 ( 0"`)F  H  0޽h ? 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O =0Multicolored Subgraphs in an Edge-colored Graph"11g0&     *Student: Yuan-Hsun Lo Advisor: Hung-Lin Fu+Z+a&   Contents a  Introduction Multicolored Subgraph Parallelism 1. Existence of MTP 2. Existence of MHCP Existence of Multicolored Subgraphs 1. Spanning Trees 2. Unicyclic Spanning Subgraphs Conclusionzlx2g)g$g2g gt  P             Preliminariesa  LA (proper) k-edge-coloring of a graph G is a mapping from E(G) into {1,& ,k } such that incident edges of G receive distinct colors. Example: 3-edge-coloring of K4 ^x  gg gggggggggggog   Preliminariesa  nA subgraph in an edge-colored graph is said to be multicolored if no two edges have the same color. Example:tox2g g3(ggc4  d   Preliminariesa  RG =A*"B is a balanced bipartite graph. Let A = { a1, a2, & , an } and B = { b1, b2, & , bn }, then the bipartite difference of aibj is defined as j  i (mod n). Example:xx xggg$ggggogggogoggggogogog ggggogoggggggggcNU  &  (  T= Preliminariesa  A k-total-coloring of a graph G is a mapping from V(G)*"E(G) into {1,& ,k } such that : (1) adjacent vertices in G receive distinct colors, (2) incident edges of G receive distinct colors, and (3) any vertex and its incident edges receive distinct colors.0xgg3 ggg ggg'gg0ggagg   Preliminariesa   Idempotent commutative latin square of order n a" n-total-colored Kn a" n-edge-colored Kn,n^xg g ggggggggoggggooogZ  %     Multicolored 1-factora  Conjecture (Transversal) In any n-edge-colored Kn,n, there exists a multicolored 1-factor for each odd integer n. Theorem (Woolbright and Fu, 1998) In any (2m-1)-edge-colored K2m where m>2, there exists a multicolored 1-factor.  x gg gggggooo<gggggg% ggggoogg+g@/  J   `  Multicolored Subgrapha&    Given an (n 1)-edge-colored Kn, a multicolored Hamiltonian path exists for n > 4. (!?) How about multicolored spanning trees?x gggggogg gg"ggg&  a ! Known Resultsa  ,Brualdi-Hollingsworth s Conjecture (1998) If m>2, then in any proper (2m 1)-edge-coloring of K2m, all edges can be partitioned into m multicolored spanning trees. Theorem (Krussel, Marshall and Verall, 2000) Three edge-disjoint multicolored spanning trees always exist.x"gge%ggggggoo$gggg$g%gg=gf B   d    M U>Multicolored Tree Parallelism ve aeae a   V?Multicolored Tree Parallelism ve aeae a  Definition K2m admits a multicolored tree parallelism (MTP) if there exists a proper (2m 1)-edge-coloring of K2m for which all edges can be partitioned into m isomorphic multicolored spanning trees. x gggoo gggggggggoo-ggg gg  " Conjecture a  Constantine s Weak Conjecture (2002) For any m>2, K2m can be (2m 1)-edge-colored in such a way that the edges can be partitioned into m multicolored isomorphic spanning trees. Constantine s Strong Conjecture (2002) If m>2, then in any proper (2m 1)-edge-coloring of K2m, all edges can be partitioned into m multicolored isomorphic spanning trees. ^xggg% ggggoo ggEggg ggggg%gg gg gg  ggoo$ggg ggJA      d  hG,Multicolored Hamiltonian Cycle Parallelism -g$ c$g$ g$g$g$c$g$ c$ - Definition K2m 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. x gggoo g*ggggggggooo-gg#g  # Known Resultsa  PConstantine s Weak Conjecture (2005) For any natural number m, K2m+1 admits an MHCP. Theorem (Constantine, 2005) If 2m+1 is an odd prime, then K2m+1 admits an MHCP.xggg%ggggooogggg%gg ggggooog  YA Conjecture a  Constantine s Strong Conjecture (2005) In any proper (2m+1)-edge-coloring of K2m+1, all edges can be partitioned into m multicolored isomorphic spanning unicyclic subgraphs. Note. Unicyclic: a graph with exactly one cycle.x2xggg%gg ggcggooo$ggg g gg5gZ          " $ Known Resultsa  fF Known Resultsa   !Multicolored Subgraph Parallelism""a&     1. Existence of MTP"g(3   MTPa  HConstatine s Weak Conjecture (2002) For any m>2, K2m can be (2m 1)-edge-colored in such a way that the edges can be partitioned into m multicolored isomorphic spanning trees. Theorem 2.1 (Constantine, 2002) If m`"1 or 3 and K2m admits an MTP, then for all r 1, admits an MTP.lx lSggg% ggggoo ggEggg ggg ggg%gg ggooggg@  4      MTPa  zTheorem 2.2 (Fu and Lo) For m`"2, K2m admits an MTP. Pf: > gg gggggoog > !Multicolored Subgraph Parallelism""a&     2. Existence of MHCP"g(3   MHCPa  Def: 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. Theorem 2.3 (Constantine, 2005) If n is an odd prime, then Kn admits an MHCP. lxlgggooo g*ggggggggooo-gggg3g ggg%ggggogg4      MHCPa  Lemma 2.4 (Fu and Lo) Let v be a composite odd integer and p be the smallest prime with p|v. Assume v = mp. If Km admits an MHCP, then Km(p) has a proper mp-edge-coloring that admits an MHCP."x g ggg ggggggggggoggoooogg#gcN[  .  4   MHCPa  PTheorem 2.5 (Fu and Lo) For each odd integer v 3, Kv admits an MHCP. Pf: Q eg egeeeemegeg&6    MHCPa  XExample: K15 Start from a 5-total-colored K5. (Consider K5(3) first.) Yx ggo$ggog o ggo g Y MHCPa   Example: K15`  ggog  (MHCPa   Example: K15`  ggog  )MHCPa   Example: K15`  ggog  * MHCPa   Example: K15`  ggog  +!MHCPa   Example: K15`  ggog  ,"MHCPa   Example: K15`  ggog  W@MHCPa  Example: K15 bx ggog  'MHCPa   Example: K15P  cckc  -#MHCPa   Example: K15P  cckc  /%MHCPa   Example: K15P  cckc  .$MHCPa   Example: K15P  cckc  0&#Existence of Multicolored Subgraphs$$c$(    1. Spanning Trees"g(3  1'MSTa  Definition  is a (2m 1)-edge-coloring of K2m, and T is a multicolored spanning tree with root x. If x is incident to two leaves e1 = xy1 and e2 = xy2, then let T[x;y1,y2;z1,z2] = T  e1  e2 + y1z1 + y2z2, for some z1, z2. eeeeeeeeeemmee+eeeeeemeemeemeem eeeeeemeemeemeemeeeemeemeememeemem eemeeme  7*MSTa  Example6ee  8+MSTa  Example6ee  6)MSTa  Theorem 3.1 (Fu and Lo) Let  be an arbitrary (2m 1)-edge-coloring of K2m and x0 be an arbitrary vertex. Then, there exist mutually edge-disjoint multicolored spanning trees, each of which contains a pendent vertex x0. ~t e eeggggggooggo+gagggog c   ZBMSTa  TNotation Tj(i) : the jth spanning tree which was constructed at round i. HU eeemmmmeem0eeeeg c @     : 4(Sketch proof of Thma&   @Round 1. Pick any one vertex x1. Let T1(1) = x1-star.Axeemeemmeeme A ;-Sketch proof of Thma&   Round 2. Pick x2, u.xeemeee  <.Sketch proof of Thma&    Round 2.  x e  =/Sketch proof of Thma&   )Round 2. Use u, u1 to construct 2nd tree.*x eeeememe * >0Sketch proof of Thma&    ?1Sketch proof of Thma&    @2Sketch proof of Thma&   Round 3. Pick x3, u.xeemeee  B3Sketch proof of Thma&   Round 3. Pick x3, u.xeemeee  :,Sketch proof of Thma&   ZDiscussion: (1) Adjust T1(n-1) ! T1(n), """""", Tn-1(n-1) ! Tn-1(n). (2) Define Tn(n) = xn-star  {"""} +{"""}. (3) Above works whenever |U| 9n-14. (4) |W| 2m-2n2+3n.jx e ggoooofggoooogoggooooo f ggooooo ggooooggogege" g  g  g  g  g  g  g  g  g  g  g  g  g  o  g  g  g @Q    S I5Sketch proof of Thma&   Discussion: We have n spanning trees if 2m 2n2 + 6n  14. Fixed 2m, we have at least spanning trees.  10 " d " d>0 " d egggggggogggg gg6gg } K6#Existence of Multicolored Subgraphs$$c$(    2. Unicyclic Spanning Subgraph"g(3@       L7MUSSa  xCorollary 3.2 (Fu and Lo) Letbe an arbitrary (2m 1)-edge-coloring of K2m 1. Then, there exist mutually edge-disjoint multicolored unicyclic spanning subgraphs (MUSS).e eegggggooo&ggg gg ggggg c Z  u         M8MUSSa  Proof. xe  N9  Conclusion"  g,3  kH Known Resultsa  cE Reference a   [1] S. Akbari, A. Alipour, H. L. Fu and Y. H. Lo, Multicolored parallelisms of isomorphic spanning trees, SIAM J. Discrete Math. 20 (2006), No. 3, 564-567. [2] R. A6. Brualdi and S. Hollingsworth, Multicolored trees in complete graphs, J. Combin. Theory Ser. B 68 (1996), No. 2, 310-313. [3] G. M. Constantine, Multicolored parallelisms of isomorphic spanning trees, Discrete Math. Theor. Compu. Sci. 5 (2002), No. 1, 121-125. [4] G. M. Constantine, Edge-disjoint isomorphic multicolored trees and cycles in complete graphs, SIAM J. Discrete Math. 18 (2005), No. 3, 577-580. [5] H. L. Fu and Y. H. Lo, Multicolored parallelisms of Hamiltonian cycles, Discrete Math. 309 (2009), No. 14, 4871-4876."xg      A      1 lI Reference a   [6] H. L. Fu and Y. H. Lo, Multicolored isomorphic spanning trees in complete graphs, Ars Combinatoria, to appear. [7] H. L. Fu and Y. H. Lo, Multicolored spanning trees in edge-colored complete graphs, SIAM J. Discrete Math., revised. [8] H. L. Fu and D. E. Woolbright, On the exists of rainbows in 1-factorizations of K2n, J. Combin. Des. 6 (1998), 1-20. [9] J. Krussel, S. Marshal and H. Verral, Spanning trees orthogonal to one-factorizations of K2n, Ars Combin. 57 (2000), 77-82. [10] H. J. 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