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Existence of MTP 2. Existence of MHCP Existence of Multicolored Subgraphs 1. Spanning Trees 2. Unicyclic Spanning Subgraphs Conclusion¡z¼²lx2g)g$g2g gªt  P            óŸ¨ Preliminaries¡a€ª  Ÿ LA (proper) k-edge-coloring of a graph G is a mapping from E(G) into {1,& ,k } such that incident edges of G receive distinct colors. Example: 3-edge-coloring of K4 ¡^†x  gg‚ÿþ gg‚gg‚gg‚gg‚ggg‚oçÿg‚ª § óŸ¨ Preliminaries¡a€ª  Ÿ¨nA subgraph in an edge-colored graph is said to be multicolored if no two edges have the same color. Example:¡tox2g€þ g‚3Ìþ(g€þg€þcª4  d  óŸ¨ Preliminaries¡a€ª  Ÿ RG =A*"B is a balanced bipartite graph. Let A = { a1, a2, & , an } and B = { b1, b2, & , bn }, then the bipartite difference of aibj is defined as j  i (mod n). Example:¡ž xx xg‚þgþg‚þ$g€þg‚þg€þg‚þo€þçÿg‚þg€þg‚þo€þçÿg‚þo‚þçÿg€þg‚þg€þg‚þo€þçÿg‚þo€þçÿg‚þo‚þçÿg‚þ g€þg‚ÿþg€þg‚ÿþo‚ÿþçÿg‚ÿþo‚ÿþçÿg€þg‚ÿþg‚þg€þg‚þg‚ÿþg€þg€þcªNU  &  (  óT=Ÿ¨ Preliminaries¡a€ª  Ÿ 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.¡0xgþg‚3Ìþ gþg‚þgþ g‚þgþg‚þ'gþg‚þ0gþg‚þagþgª  óŸ¨ Preliminaries¡a€ª  Ÿ º Idempotent commutative latin square of order n a" n-total-colored Kn a" n-edge-colored Kn,n¡®^xg gÿþ gÿþgg‚gg€g‚g€g‚o‚çÿg€g‚g€g‚o‚çÿo€çÿo‚çÿg‚ªZ  %     óŸ¨Multicolored 1-factor¡a€ª  Ÿ¨æConjecture (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. ¡ çx g�g gÿ™þgg‚gg‚o‚çÿoçÿo‚çÿ<gg‚gg�gg˜›%þ gg‚gg‚oçÿo‚çÿgg‚+gª@/  J   ` ó Ÿ¨Multicolored Subgraph¡a€ª&    Ÿ üGiven an (n 1)-edge-colored Kn, a multicolored Hamiltonian path exists for n > 4. (!?) How about multicolored spanning trees?¡x gg‚g€gg‚o‚çÿgg gg‚"ggÿþgª&  a ó!Ÿ¨ Known Results¡a€ª  Ÿ ,Brualdi-Hollingsworth s Conjecture (1998) If m>2, then in any proper (2m ÿ1)-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"g�ge˜›%þgg‚gg‚€g€g‚o€çÿo‚çÿ$g€g‚g€g�$g€˜›%þg€g€ÿþ=g€ªf B   d    M óU>Ÿ¨Multicolored Tree Parallelism ¡ve€ÿþ a€e€ÿþa€e€ÿþ a€ª  Ÿª óV?Ÿ¨Multicolored Tree Parallelism ¡ve€ÿþ a€e€ÿþa€e€ÿþ a€ª  Ÿ ’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. ¡ÊÊx g�gg‚oçÿo‚çÿ gg‚gƒggÿþgg‚gg‚oçÿo‚çÿ-gg‚g gÿþgª Ê ó"Ÿ¨ Conjecture¡ a€ª  Ÿ ºConstantine s Weak Conjecture (2002) For any m>2, K2m can be (2m ÿ1)-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 ÿ1)-edge-coloring of K2m, all edges can be partitioned into m multicolored isomorphic spanning trees.¡ ^xg�gg˜›%þ gg‚gg‚oçÿo‚çÿ gg‚€Eg€g‚g€ g€ÿþg€g�g€g€˜›%þg€g‚ g€g€ÿþ g€g‚ € g€g‚o€çÿo‚çÿ$g€g‚g€ g€ÿþg€ªJA   ¶   d  óhGŸ¨,Multicolored Hamiltonian Cycle Parallelism ¡Ä-g€$ÿþ c€$g€$ÿþ g€$g€$ÿþg€$c€$g€$ÿþ c€$ª - Ÿ¨ÒDefinition 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. ¡¸Óx g�gg‚oçÿo‚çÿ g*g‚gƒggÿþgg‚gg‚oçÿo‚çÿoçÿ-gg‚#gª Ó ó#Ÿ¨ Known Results¡a€ª  Ÿ 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.¡©xg�gg˜›%þgg‚gg‚oçÿo‚çÿoçÿgg�gg˜›%þgg‚ ggÿþgg‚oçÿo‚çÿoçÿgª © óYAŸ¨ Conjecture¡ a€ª  Ÿ ÂConstantine 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.¡Ê°x2xg�gg˜›%þggÿþ gg‚c€g€g‚o€çÿo‚çÿo€çÿ$g€g‚g€ g€ÿþ g€g€ÿþ5g€ªZš          " ó$Ÿ¨ Known Results¡a€ª  ófFŸ¨ Known Results¡a€ª  ó Ÿ¨!Multicolored Subgraph Parallelism¡""a€ª&     Ÿ¨1. Existence of MTP¡"g(3þª  ó Ÿ¨MTP¡a€ª  Ÿ HConstatine s Weak Conjecture (2002) For any m>2, K2m can be (2m ÿ1)-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.¡´²’lx ’lSg�gg˜›%þ gg‚gg‚oçÿo‚çÿ gg‚€Eg€g‚g€ g€ÿþg€g€þ g�þg€þg€˜›%þg€þg‚þ g€þg‚þo€þçÿo‚þçÿg€þg‚þg€þª@  4   ã  ó Ÿ¨MTP¡a€ª  Ÿ zTheorem 2.2 (Fu and Lo) For m`"2, K2m admits an MTP. Pf: ¡à> g�g ggg‚gg‚oçÿo‚çÿgª > óŸ¨!Multicolored Subgraph Parallelism¡""a€ª&     Ÿ¨2. Existence of MHCP¡"g(3þª  ó Ÿ¨MHCP¡a€ª  Ÿ¨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. ¡Þ’lx’lg�þgþg‚þoþçÿo‚þçÿoþçÿ gþ*g‚þgƒþgþgÿþgþg‚þgþg‚þoþçÿo‚þçÿoþçÿ-gþg‚þgþg3Ìþgþ g�þgþg˜›%þgþg‚þgþg‚þo‚þçÿgþgþª4     ó Ÿ¨MHCP¡a€ª  Ÿ¨ÄLemma 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.¡"Åx g�þ ggþg‚þ gþg‚þgþg‚þgþg‚þgþg‚þgþg‚þo‚þçÿgþg‚þo‚þçÿoþçÿo‚þçÿoþçÿgþg‚þ#gþc€ªN[  .  4  ó Ÿ¨MHCP¡a€ª  Ÿ¨PTheorem 2.5 (Fu and Lo) For each odd integer v 3, Kv admits an MHCP. Pf: ¡Q e�g egee‚ee‚m‚çÿegegª&6   ó Ÿ¨MHCP¡a€ª  Ÿ¨XExample: K15 Start from a 5-total-colored K5. (Consider K5(3) first.)¡ Yx gg‚oçÿ$gg‚oçÿg oçÿ gg‚oçÿ gª Y óŸ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó(Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó)Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó* Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó+!Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó,"Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  óW@Ÿ¨MHCP¡a€ª  Ÿ¨Example: K15 ¡bx gg‚oçÿgª  ó'Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡P  cc‚kçÿcª  ó-#Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡P  cc‚kçÿcª  ó/%Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡P  cc‚kçÿcª  ó.$Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡P  cc‚kçÿcª  ó0&Ÿ¨#Existence of Multicolored Subgraphs¡$$c€$ª(    Ÿ¨1. Spanning Trees¡"g(3þª  ó1'Ÿ¨MST¡a€ª  Ÿ ÄDefinition Æ 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.¡¾ã‚ e�e‚e€e‚e€e‚e€e‚e€e‚m€çÿm‚çÿe€e‚+e€e‚e€e‚e€e‚m€çÿe€e‚m€çÿe€e‚m€çÿe€e‚m€çÿ e€e‚ÿþe€ÿþe‚ÿþe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe‚ÿþm€ÿþçÿ e€e‚m€çÿe€e‚m€çÿe€ª ã ó7*Ÿ¨MST¡a€ª  Ÿ¨Example¡6‚e�e€ª  ó8+Ÿ¨MST¡a€ª  Ÿ¨Example¡6‚e�e€ª  ó6)Ÿ¨MST¡a€ª  Ÿ äTheorem 3.1 (Fu and Lo) Let Æ 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. ¡ð~‚t–‚ e� eegg‚g€g‚g€g‚o€çÿo‚çÿg€g‚o€çÿ+g€ag€g€ÿþg‚ÿþo€ÿþçÿg€ c€ ª ó óZBŸ¨MST¡a€ª  Ÿ¨TNotation Tj(i) : the jth spanning tree which was constructed at round i. ¡HU‚ e�ee‚ÿþm‚ÿþçÿmÿþm‚ÿþmÿþee‚ÿþmÿþ0eeÿþe‚ÿþeg€ c€ ª@     : ó4(Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨@Round 1. Pick any one vertex x1. Let T1(1) = x1-star.¡âAxee‚mçÿee‚mçÿmee‚mçÿeª A ó;-Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨Round 2. Pick x2, u.¡€xee‚mçÿee‚eª  ó<.Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨ Round 2. ¡ x eª  ó=/Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨)Round 2. Use u, u1 to construct 2nd tree.¡¦*x ee‚ee‚mçÿemeª * ó>0Ÿ¨Sketch proof of Thm¡a€ª&   Ÿª ó?1Ÿ¨Sketch proof of Thm¡a€ª&   Ÿª ó@2Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨Round 3. Pick x3, u.¡€xee‚mçÿee‚eª  óB3Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨Round 3. Pick x3, u.¡€xee‚mçÿee‚eª  ó:,Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ ZDiscussion: (1) Adjust T1(n-1) ’! T1(n), Å"Å"Å"Å"Å"Å", Tn-1(n-1) ’! Tn-1(n). (2) Define Tn(n) = xn-star  {Å"Å"Å"} +{Å"Å"Å"}. (3) Above works whenever |U| 9n-14. (4) |W| 2m-2n2+3n.¡j®x e gg‚oçÿoo‚of€g€g‚o€çÿo€o‚o€g€o€g€g‚o‚çÿo€çÿo€o‚o€ f€ g€g‚o‚çÿo€çÿo€o‚o€ g€g‚o‚çÿo€o‚o€g€g‚o‚çÿg€e€g€e€" g€  g€ ÿþ g‚ ÿþ g€ ÿþ g‚ ÿþ g€ ÿþ g€  g€ ÿþ g‚ ÿþ g€ ÿþ g‚ ÿþ g€ ÿþ g‚ ÿþ o€ ÿþ g€ ÿþ g‚ ÿþ g‚ ª@Q    S óI5Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ øDiscussion: We have n spanning trees if 2m 2n2 + 6n  14. Fixed 2m, we have at least spanning trees. ¡Î ‚1ÿ0 " dÿ‚ÿ " dÿ>ÿ0 " dÿ‚‚ egg‚ggÿþg‚ÿþgÿþg‚ÿþoÿþgÿþg‚ÿþg€ÿþgÿþ gg‚6ggÿþª } óK6Ÿ¨#Existence of Multicolored Subgraphs¡$$c€$ª(    Ÿ¨2. Unicyclic Spanning Subgraph¡"g(3þª@       óL7Ÿ¨MUSS¡a€ª  Ÿ xCorollary 3.2 (Fu and Lo) LetÆbe an arbitrary (2m 1)-edge-coloring of K2m 1. Then, there exist mutually edge-disjoint multicolored unicyclic spanning subgraphs (MUSS).¡ö‚£–e� eegg€g‚g€g‚o€çÿo‚çÿo€çÿ&g€g€g€ÿþ g€g€ÿþ g€g€ÿþg€g€ÿþg€ c€ ªZ  u         óM8Ÿ¨MUSS¡a€ª  Ÿ¨Proof.¡ xeª  óN9Ÿª Ÿ¨ Conclusion¡"  g,3þª  ókHŸ¨ Known Results¡a€ª  ócEŸ¨ Reference¡ a€ª   Ÿ¨¼[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.¡"½x½gª¨    �  A  Š    1 ólIŸ¨ Reference¡ a€ª   Ÿ¨‹[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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ð6…‡�ƒ¿Àÿˆ"ñ¿`ð9 ™€ ðt2 ðl “ ð6…‡�ƒ¿Àÿˆ"ñ¿`ð‰ é€ ðt2 ðl “ ð6…‡�ƒ¿Àÿˆ"ñ¿`ðt  Ô ð„B ðl ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿð ` 6 ð„B ðlB ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðp` % ð„B ðl ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿð 2ÎS ð„B ðl ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðp% pÎ ðú¢ ðl ó ðZ€lRò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð.Î lî  ðhŸ¨x1¡6 2emçÿª  ðú¢ ðl ó ðZ€ÌUò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�"¿B ðhŸ¨x2¡6 2emçÿª  ðú¢ ðl ó ðZ€Zò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðè Î &î  ðhŸ¨x3¡6 2emçÿª  ð„B ðl ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿð©; ©ä ðú¢ ðl ó ðZ€°^ò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðsC±c  ðhŸ¨x4¡6 2emçÿª  ðú¢ ðl ó ðZ€Lbò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð` `ž€  ðhŸ¨x5¡6 2emçÿª  ðú¢ ðl ó ðZ€¼fò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�& ¿F  ðhŸ¨x6¡6 2emçÿª  ðþ¢ ðl ó ðZ€”kò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðhd FÑ ðlŸ¨T1¡: 2g o çÿª  ð1¢ ð l ó ðZ€¬oò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð]ù¸ ¤ 𧟨Ex. 5-edge-colored K6¡b 2ggoçÿgª  ðH ðl ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ–ŠŽº___PPT10‹në.�Ë è­­+Y ÛDñB'ñ mÿÿÿÿ=ñ @Bñ Dñý'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*lÿÿÿÿÿÿÿÿ%ñ(ñ+î§ï !€ 0 ¾ð¶ððpðNð( ð ðpðŠ ðp “ ð6€¬yò‡�¿Àÿ ¬ˆð,ù„Sðà  ò ð žðø¢ ðp ó ðZ€äzò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�9=M ðnŸ¨ Introduction¡$ 2 €g€ª   ð„ ðp ƒ ð0€$~ò�¿Àÿ ¬ˆðmùâ“ ðà ò ð žðH ðp ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.�Ë è­­+Dñ='ñ mÿÿÿÿ=ñ @Bñ +î›ï !€ 0 ²ðªðtðBð( ð ðtðx ðt c ð$€ �ò¿ÿ¬ˆðÕùYiðà ò ð žðŠ ðt “ ð6€Ð�ò‡�¿Àÿ ¬ˆð,ù Sðà  ò ð žðø¢ ðt ó ðZ€�ò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�9=M ðnŸ¨ Introduction¡$ 2 €g€ª   ðH ðt ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ºË þ]+Dñ='ñ mÿÿÿÿ=ñ @Bñ +î§ï !€ 0 ¾ð¶ðð°ðNð( ð ð°ðŠ ð° “ ð6€àQt‡�¿Àÿ ¬ˆð,ù„Sðà  t ð žðø¢ ð° ó ðZ€¼×t¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�9=M ðnŸ¨ Introduction¡$ 2 €g€ª   ð„ ð° ƒ ð0€ün:�¿Àÿ ¬ˆðõùâðà t ð žðH ð° ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.�Ë è­­+Dñ='ñ mÿÿÿÿ=ñ @Bñ +îÌï !€ 0 ÞðÖ ðxðnð( ð ðxð¤ ðx c ð$€´ ò¿ÿ¬ˆðÕùâið<$ñD 0à ò ð žðŠ ðx “ ð6€x¡ò‡�¿Àÿ ¬ˆð,ù Sðà  ò ð žðø¢ ðx ó ðZ€Ä¢ò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�9=M ðnŸ¨ Introduction¡$ 2 €g€ª   ðH ðx ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ–ŠŽº___PPT10‹në.ºË þ]+ ¶˜CDñB'ñ mÿÿÿÿ=ñ @Bñ Dñý'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*xX©%ñ(ñ+îÌï !€ 0 ÞðÖ@ð„ðnð( ð ð„ð¤ ð„ c ð$€˜Õò¿ÿ¬ˆðÕùÿÑð<$ñD 0à ò ð žðŠ ð„ “ ð6€\Öò‡�¿Àÿ ¬ˆð,ù Sðà  ò ð žðø¢ ð„ ó ðZ€¨×ò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�9=M ðnŸ¨ Introduction¡$ 2 €g€ª   ðH ð„ ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ–ŠŽº___PPT10‹në.ºË þ]+ ¶˜CDñB'ñ mÿÿÿÿ=ñ @Bñ Dñý'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*„°â%ñ(ñ+î™ï !€ 0 °ð¨@ð£€ð@ð( ð ð€ð� ð€ £ ð<€ ëò‡�¿Àÿ ¬ˆð,ùYSðà  ò ð žðø¢ ð€ ó ðZ€€ìò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�9=M ðnŸ¨ Introduction¡$ 2 €g€ª   ðpð„ ðùÕY; 𣀠ð#"ñŸ Ã]ÅõðÕùY;ðà òð# ð€ c ð„€xùò��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðAF Y; ðgŸ¨?¡ g0ÿþª  ¦ø @`€ð# ð€ c ð„€„µ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ð) F A; ðgŸ¨?¡ g0ÿþª  ¦ø @`€ðz ð€ c ð„€µ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðF ) ; 𾟨2m+1=prime (Constantine)¡^gÿþg‚ÿþ gÿþgff3þª  ¦ø @`€ð ð€ c ð„€` µ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðùF ; ðFŸ¨ K2m+1 (spanning unicyclic)¡Èeg‚$o$çÿo‚$çÿo$çÿg$ g gÿþgª&    ¦ø @`€ð# ð € c ð„€øµ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðA�YF  ðgŸ¨?¡ g0ÿþª  ¦ø @`€ðÂ ð € c ð„€`,µ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ð) �AF  🨠3 (Krussel, et.al.)¡rggÿþggff3þg€ff3þª@      ¦ø @`€ðÎ ð € c ð„€ÄFµ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ð�) F  ðŸ 8m=2k or 5Å"2k (Constantine)¡”g‚ÿþg€ÿþo‚ÿþg€ÿþo‚ÿþg€ff3þª  ¦ø @`€ð¦ ð € c ð„€XTµ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðù�F  ð꟨ K2m (spanning tree)¡Žg g‚$o$çÿo‚$çÿg‚$gª  ¦ø @`€ðG ð € c ð„€Mµ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðAÕY� ð‹Ÿ¨ Given (Isomorphic)¡2e gÿþª  ¦ø @`€ð: ð€ c ð„€Œcµ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ð) ÕA� ð~Ÿ¨Given ¡2egª  ¦ø @`€ðJ ð€ c ð„€ìkµ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðÕ) � ðŽŸ¨Assigned (Isomorphic)¡2 egÿþª  ¦ø @`€ð8 ð€ c ð„€Ôwµ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðùÕ� ð|Ÿ¨ coloring ¡ gª  ¦ø @`€ð„B ð€ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðùÕYÕð~B ð€ Ó ðN…¿ÀÂËœ1ÒÓÔÕÿ ?¿ðùF YF ð„B ð€ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðù;Y;ð„B ð€ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðùÕù;ð~B ð€ Ó ðN…¿ÀÂËœ1ÒÓÔÕÿ ?¿ð) Õ) ;ð~B ð€ Ó ðN…¿ÀÂËœ1ÒÓÔÕÿ ?¿ðAÕA;ð„B ð€ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðYÕY;ð„B ð€ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðÕ;ð„B ð€ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðù�Y�ð„B 𛀠ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðùÕ�ðH ð€ ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ºË þ]+Dñ='ñ mÿÿÿÿ=ñ @Bñ +î6ï !€ 0 MðEÐð¨ðÝð( ð ð¨ð� 𨠣 ð<€ö‡�¿Àÿ ¬ˆð,ùYSðà  ö ð žðø¢ ð¨ ó ðZ€˜ö¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�9=M ðnŸ¨ Introduction¡$ 2 €g€ª   ð5ðŠ ðùÕY; ð¨# ð ˆ#"ñŸ Ã]ÅõðÕùY;ðà öð# ð¨ c ð„€À"ö��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðAF Y; ðgŸ¨2¡ gÿþª  ¦ø @`€ð ð¨ c ð„€´¥t��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ð) F A; ð^Ÿ¡ g0ÿþª  ¦ø @`€ð ð¨ c ð„€$çt��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðF ) ; ð^Ÿ¡ gff3þª  ¦ø @`€ð ð¨ c ð„€œªt��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðùF ; ðFŸ¨ K2m+1 (spanning unicyclic)¡Èeg‚$o$çÿo‚$çÿo$çÿg$ g gÿþgª&    ¦ø @`€ðƒ ð ¨ c ð„€€˜t��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðA�YF  ðÇŸ¨3 trees if m>13¡rgÿþgg‚ggÿþª  ¦ø @`€ð< ð ¨ c ð„€h¥:��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ð) �AF  ð€Ÿ¨ ¡6gg€ff3þª  ¦ø @`€ð ð ¨ c ð„€Ð²:��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ð�) F  ð`Ÿ¡"g€ff3þª  ¦ø @`€ð¦ ð ¨ c ð„€ÔÎ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðù�F  ð꟨ K2m (spanning tree)¡Žg g‚$o$çÿo‚$çÿg‚$gª  ¦ø @`€ðG ð ¨ c ð„€œG,��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðAÕY� ð‹Ÿ¨ Given (Isomorphic)¡2e gÿþª  ¦ø @`€ð: ð¨ c ð„€˜:��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ð) ÕA� ð~Ÿ¨Given ¡2egª  ¦ø @`€ðJ ð¨ c ð„€ŒN(��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðÕ) � ðŽŸ¨Assigned (Isomorphic)¡2 egÿþª  ¦ø @`€ð8 ð¨ c ð„€81ö��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ?ðùÕ� ð|Ÿ¨ coloring ¡ gª  ¦ø @`€ð„B ð¨ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðùÕYÕð~B ð¨ Ó ðN…¿ÀÂËœ1ÒÓÔÕÿ ?¿ðùF YF ð„B ð¨ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðù;Y;ð„B ð¨ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðùÕù;ð~B ð¨ Ó ðN…¿ÀÂËœ1ÒÓÔÕÿ ?¿ð) Õ) ;ð~B ð¨ Ó ðN…¿ÀÂËœ1ÒÓÔÕÿ ?¿ðAÕA;ð„B ð¨ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðYÕY;ð„B ð¨ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðÕ;ð„B ð¨ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðù�Y�ð„B ð¨ ã ðT…¿ÀÂËŸoÒÓÔÕ×ÿ ?¿ðùÕ�ðÞ¢ ð¨ ó ðZ€DÃt¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðp‡Õ w  ðTŸ Ç¡  2f0ÿþª ðÞ¢ ð¨ ó ðZ€˜z:¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðB ‡Õ I ðTŸ Ç¡  2f0ÿþª ð~² 𨠳 ðBA Ž?�ƒ¿Àÿ ?ð— x ð Á Žð~² 𨠳 ðBA �?�ƒ¿Àÿ ?ð�h + ð Á �ðH 𨠃 ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ºË þ]+Dñ='ñ mÿÿÿÿ=ñ @Bñ +î‰ï !€ 0  ð˜Ðð$ð0ð( ð ð$ðx ð$ c ð$€º¿ÿ¬ˆð,ùYSðà  º ð žðx ð$ c ð$€¸.º¿ÿ¬ˆð»ùY€ ðà º ð žðH ð$ ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ÇË þ~+Dñ='ñ mÿÿÿÿ=ñ @Bñ +îý ï !€ 0 kðcàð (ðûð( ð ð(ðž ð( S ð€”/º¿ÿ¬ðÕùY´ð<$ñ@ 0à º ð žðŠ ð( “ ð6€Ì(º‡�¿Àÿ ¬ˆð,ù Sðà  º ð žð'¢ ð( ó ðZ€81º¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      𪲠ð( ³ ðBA ?�ƒ¿Àÿ ?ð¡ ïáð8Á $ñ@ 0𪲠ð( ³ ðBA ?�ƒ¿Àÿ ?ðÝ óžª ð8Á $ñD 0ðH ð( ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ:Š2º___PPT10‹ë.ÇË0×8ä+8mäDñæ'ñ mÿÿÿÿ=ñ @Bñ Dñ¡'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +DñØ'ñ =ñ%ñ(ñÿÿÿÿDñ€'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*(²Ò%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*(Ò%%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*(ÿÿÿÿÿÿÿÿ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*(ÿÿÿÿÿÿÿÿ%ñ(ñ+îéï !€ 0 ðøðð ,ð�ð( ð ð,ð� ð, £ ð<€ÜFº�ƒ¿Àÿ ¬ˆðÕÌYPðà º ð žðÿ¢ ð, ƒ ð0€\Hº¿�ƒ¿Àÿð¦þäŽ ðHˆ@Š8º___PPT9‹¬ ðOŸ¨cProve the case that m is an odd integer. By Theorem 2.1 and Lemma 2.2 , we can conclude the proof.¡¦d% þ2ggg gg ggª d ¦øØø8XðŠ ð , “ ð6€¸Mº‡�¿Àÿ ¬ˆð,ù Sðà  º ð žð'¢ ð , ó ðZ€ðNº¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      ðH ð, ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ÊË`c–r+Dñ='ñ mÿÿÿÿ=ñ @Bñ +î}ï !€ 0 ”ðŒÀð ð$ð( ð ð ðr ð  S ð€ìbº¿ÿ¬ð,ùYSðà  º ð žðr ð  S ð€Äcº¿ÿ¬ð»ùY€ ðà º ð žðH ð  ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ÇË þ~+Dñ='ñ mÿÿÿÿ=ñ @Bñ +îõï !€ 0 ðÿ�ðð—ð( ð ððž ð S ð€¨iº¿ÿ¬ðÕùY´ð<$ñD 0à º ð žðŠ ð “ ð6€Pjº‡�¿Àÿ ¬ˆð,ù Sðà  º ð žð'¢ ð ó ðZ€œkº¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ–ŠŽº___PPT10‹në.ÁË@ ôo+Ÿ—ÉYDñB'ñ mÿÿÿÿ=ñ @Bñ Dñý'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*Ï%ñ(ñ+îÄï !€ 0 ÛðÓ ððkð( ð ððr ð S ð€¼wº¿ÿ¬ðÕùY´ðà º ð žðŠ ð “ ð6€ Œ,‡�¿Àÿ ¬ˆð,ù Sðà  º ð žð'¢ ð ó ðZ€h€º¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ÁË`;Lp+Dñ='ñ mÿÿÿÿ=ñ @Bñ +îAï !€ 0 XðP°ððèð( ð ðð� ð  £ ð<€ˆ,‡�¿Àÿ ¬ˆð,ùYSðà  º ð žð– ð  ³ ðB€\ù,�ƒ¿Àÿ ¬ˆðÕù,øðà , ð žð—¢ ð  ƒ ð0€¬�º¿�ƒ¿Àÿð¦þä� ðHˆ@Š8º___PPT9‹¬ ð矨%By induction on v. Using Lemma 2.4. ¡|&% þ2gg g ggª & ¦ø @`€ ð'¢ ð  ó ðZ€ žº¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      𬲠ð0  ð`A ?�ƒ†Á¿ÀÅÁÿ ?¬ˆð@{  ðÁ à ºðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ÁË5kp+Dñ='ñ mÿÿÿÿ=ñ @Bñ +îtï !€ 0 ¨ð ð#'0ð8ð( ð ð0ðž ð0 S ð€líº¿ÿ¬ðÕù¡S ð<$ñD  0à º ð žðŠ ð0 “ ð6€îº‡�¿Àÿ ¬ˆð,ù Sðà  º ð žð'¢ ð0 ó ðZ€ŒÅò¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      ð™ðl ð�›ž, ð'0ð›�ž,ð,$ñD 0ðTB ð 0B c ð$D¿ËjJÿðFS  t ðTB ð 0 c ð$D¿ËjJÿðFt V wðTB ð 0 c ð$D¿ËjJÿðV w wðTB ð 0‚ c ð$D¿ËjJÿð t ò wðTB ð 0 c ð$D¿ËjJÿð S ò t ðTB ð0B c ð$D¿ËjJÿðV S  wðTB ð0 c ð$D¿ËjJÿð S  wðTB ð0 c ð$D¿ËjJÿðFt  wðTB ð0 c ð$D¿ËjJÿðFt ò t ðTB ð0‚ c ð$D¿ËjJÿðV t ò w𻢠ð0 ƒ ð0€üﺿ�ƒ¿ÀÿðÁ Xw , ðSŸ¨1¡" 2gþª  𻢠ð0 ƒ ð0€lôº¿�ƒ¿ÀÿðÁ Í w ¡  ðSŸ¨1¡" 2gþª  𻢠ð0 ƒ ð0€øº¿�ƒ¿Àÿð ˜  l  ðSŸ¨2¡" 2gþª  𻢠ð0 ƒ ð0€Äûº¿�ƒ¿Àÿðc � a  ðSŸ¨2¡" 2gþª  𻢠ð0 ƒ ð0€ÌÏ¿�ƒ¿Àÿðö œ ¬ p  ðSŸ¨3¡" 2gþª  𻢠ð0 ƒ ð0€ Ï¿�ƒ¿Àÿð, ® â ‚  ðSŸ¨3¡" 2gþª  𻢠ð0 ƒ ð0€¸Ï¿�ƒ¿ÀÿðI 8 ÿ  ðSŸ¨4¡" 2gþª  𻢠ð0 ƒ ð0€d Ï¿�ƒ¿Àÿð. À ä ”  ðSŸ¨4¡" 2gþª  𻢠ð0 ƒ ð0€8Ï¿�ƒ¿Àÿðø ƒ ® W  ðSŸ¨5¡" 2gþª  𻢠ð0 ƒ ð0€è Ï¿�ƒ¿Àÿð~ ÏR  ðSŸ¨5¡" 2gþª  ð˜2 ð0 ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðÁ žw T ð˜2 ð0 ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð¼[r ð˜2 ð0 ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðô dª ð˜2 ð 0 ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð‘ý G³ ð˜2 ð!0 ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðè ë ž¡ 𻢠ð"0 ƒ ð0€Ï¿�ƒ¿ÀÿðÊ ›€ o  ðSŸ¨1¡" 2gþª  𻢠ð#0 ƒ ð0€(Ï¿�ƒ¿Àÿð�ý FÑ  ðSŸ¨2¡" 2gþª  𻢠ð$0 ƒ ð0€ÔÏ¿�ƒ¿Àÿð¼Xr , ðSŸ¨3¡" 2gþª  𻢠ð%0 ƒ ð0€€"Ï¿�ƒ¿Àÿðü X² , ðSŸ¨4¡" 2gþª  𻢠ð&0 ƒ ð0€&Ï¿�ƒ¿Àÿðà ë –¿  ðSŸ¨5¡" 2gþª  ðH ð0 ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆt Šl º___PPT10‹L ë.ÎË0%u+ä·¦èDñ 'ñ mÿÿÿÿ=ñ @Bñ DñÛ'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñç'ñ =ñ%ñ(ñÿÿÿÿDñ�'ñ =ñ%ñ(ñDñ7'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*06%ñ(ñDñ«'ñ ô=ñ-ñs6ñBñwipe(down)*ñ<3ñ<ñû*06Dñç'ñ =ñ%ñ(ñÿÿÿÿDñ�'ñ =ñ%ñ(ñDñ7'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*06Y%ñ(ñDñ«'ñ ô=ñ-ñs6ñBñwipe(down)*ñ<3ñ<ñû*06YDñ4'ñ =ñ%ñ(ñÿÿÿÿDñÜ'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*'0ÿÿÿÿÿÿÿÿ%ñ(ñ+îc~ï !€ 0 ûCðóC€ð_zPð‹Cð( ð ðPðŠ ðP “ ð6€L7χ�¿Àÿ ¬ˆð,ù Sðà  Ï ð žð'¢ ðP ó ðZ€Ô2Ï¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      ð„ ðP ƒ ð0€¨4Ï�¿Àÿ ¬ˆð–:·ðà Ï ð žð»ð8 ð å—¥ ðkPðå —¥𻢠ðP ƒ ð0€ø;Ï¿�ƒ¿Àÿð£åY¹ ðSŸ¨1¡" 2gþª  ð˜2 ðP ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð£éYŸð˜2 ðP ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð£`Y𻢠ð P ƒ ð0€Ì?Ï¿�ƒ¿Àÿð¢`X4 ðSŸ¨2¡" 2gþª  ð˜2 ð P ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð£Þ Y” 𻢠ð P ƒ ð0€xCÏ¿�ƒ¿Àÿð£Û Y¯  ðSŸ¨3¡" 2gþª  ð˜2 ð P ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð¤Y Z 𻢠ð P ƒ ð0€pGÏ¿�ƒ¿Àÿð£V Y*  ðSŸ¨4¡" 2gþª  ð˜2 ðP ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð£ÑY‡𻢠ðP ƒ ð0€�KÏ¿�ƒ¿Àÿð›ÑQ¥ ðSŸ¨5¡" 2gþª  ð„B ðP ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÿšÿ`ð„B ðP ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÿÿÛ ð„B ðP ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÿ� ÿV ð„B ðP ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÿ ÿÑðê ðP c ðº…B¶CöDEÁFÁ�ƒ†A¿ÀÂÅAËœ1ÒÓÔÕ×ÿðÿ[ê¶Ô¶¨¶|”ö@­ ­ ¬€ðY@6 ðð ðP s ðÀ…B¶CöDEÁFÁ�ƒ†Á¿ÀÂÅÁËœ1ÒÓÔÕ×ÿˆðÿ[ê¶Ô¶¨¶|”ö@­ ­ ¬€ðY6 ,ðð ðP s ðÀ…B¶CöDEÁFÁ�ƒ†Á¿ÀÂÅÁËœ1ÒÓÔÕ×ÿˆðÿ[ê¶Ô¶¨¶|”ö@­ ­ ¬€ðY»± ðê ðP c 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2gþª  𻢠𬔠ƒ ð0€pJп�ƒ¿ÀÿðáV —*  ðSŸ¨4¡" 2gþª  𻢠ð­” ƒ ð0€Tп�ƒ¿ÀÿðáÛ —¯  ðSŸ¨3¡" 2gþª  𻢠ð®” ƒ ð0€„Yп�ƒ¿Àÿð ê ¾  ðSŸ¨3¡" 2gþª  𻢠𯔠ƒ ð0€|\п�ƒ¿ÀÿðÑ¢ ‡v ðSŸ¨2¡" 2gþª  𻢠ð°” ƒ ð0€àYп�ƒ¿ÀÿðÑ' ‡û  ðSŸ¨1¡" 2gþª  𻢠ð±” ƒ ð0€€cп�ƒ¿ÀÿðѬ‡€  ðSŸ¨5¡" 2gþª  𻢠ð²” ƒ ð0€ôhп�ƒ¿ÀÿðÁQw%  ðSŸ¨5¡" 2gþª  𻢠ð³” ƒ ð0€älп�ƒ¿ÀÿðÁ¯ wƒ  ðSŸ¨1¡" 2gþª  ðÃðl ðÑ ‡Î  ð·”ðš ¿`ð,$ñD 0ð„B ð´” ã ðT…D¿ÀÿÂË8cÒÓÔÕ×ÿðÿ ÿÎ 𻢠ðµ” ƒ ð0€Ôpп�ƒ¿ÀÿðѾ ‡’  ðSŸ¨4¡" 2gÿþª  ðÄð¦ ð:å= ðÞ”# ð ˆ#"ñŸ Ãðå:=ð,$ñD 0ð& ðß” c ð„€DwÐ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ ?ð<= ðjŸ¨12¡"gª  ¦ø @`€ð& ðà” c ð„€h€Ð��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ ?ð;< ðjŸ¨10¡"gª  ¦ø @`€ð& ðá” c ð„€à‰Ð��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ ?ð:; ðjŸ¨11¡"gª  ¦ø @`€ð& ðâ” c ð„€xšÐ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ ?ð<õ= ðjŸ¨10¡"gª  ¦ø @`€ð& ðã” c ð„€ì£Ð��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ ?ð;õ< ðjŸ¨11¡"gª  ¦ø @`€ð& ðä” c ð„€´žÐ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ ?ð:õ; ðjŸ¨12¡"gª  ¦ø @`€ð& ðå” c ð„€<¸Ð��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ ?ð<å=õ ðjŸ¨11¡"gª  ¦ø @`€ð& ðæ” c ð„€ ÀÐ��_‚жƒ�_„ж‡¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿ ?ð;å<õ ðjŸ¨12¡"gª  ¦ø @`€ð& ðç” c 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=ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*‹œÿÿÿÿÿÿÿÿ%ñ(ñDñ‚'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*§œÿÿÿÿÿÿÿÿ%ñ(ñDñ÷'ñ =ñ%ñ(ñÿÿÿÿDñŸ'ñ =ñ%ñ(ñDñG'ñ =ñ4@Bñ�Bñ Bñ °Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*îœÿÿÿÿÿÿÿÿ%ñ(ñDñ»'ñ ô=ñ-ñƒ6ñBñ'blinds(horizontal)*ñ<3ñ<ñû*îœÿÿÿÿÿÿÿÿ+îü>ï !€ 0 ß4ð×4ððN•¬ñðc4ð( ð ð¬ðŠ ð¬ “ ð6€ÖÓ‡�¿Àÿ ¬ˆð,ù Sðà  Ó ð žð'¢ ð¬ ó ðZ€T×Ó¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      ð„ 𬠃 ð0€TÜÓ�¿Àÿ ¬ˆð–:·ðà Ó ð žðŠ2 ð¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð«  mðŠ2 ð¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð�«  èðŠ2 ð¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð «  c ðŠ2 ð¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðƒ «  Þ ðŠ2 ð ¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðþ«  YðŠ2 ð ¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð dmðŠ2 ð ¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð� dèðŠ2 ð ¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð dc ðŠ2 ð ¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðƒ dÞ ðŠ2 ð¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðþ dYðŠ2 ð¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðfÁmðŠ2 ð¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð�fÁèðŠ2 ð¬ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð fÁc ðŠ2 ð¬ ã 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ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿð= V ¨ V ðü¢ ð|¬ ó ðZ€è Õ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðt¢ v ðjŸ¨Bipartite difference = 0¡" 2gÿþª  ðŠB ð}¬ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿð= þ ¨ þ ðü¢ ð~¬ ó ðZ€¸Õ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðt• i ðjŸ¨Bipartite difference = 2¡" 2gÿþª  ðŠB ð¬ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿð= ñ¨ ñð(¢ 𑬠ó ðZ€èÕ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð†®t¦ð,$ñ 0 ðjŸ¨C(1)¡6 2emª  ð)ðl ðõ mwþ ð•¬ðmõ wþð,$ñD 0ðŠB ðI¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆðØ mØ �ðŠB ðJ¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆð6m6�ðŠB ðK¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆð”m”�ðŠB ðR¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆðØ èØ  ðŠB ðS¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆð6è6 ðŠB ðT¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆð”è” ðŠB ð[¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆðØ c Ø ƒ ðŠB ð\¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆð6c 6ƒ ðŠB ð]¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆð”c ”ƒ ðŠB ðd¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆðØ Þ Ø þðŠB ðe¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆð6Þ 6þðŠB ðf¬ ó ðZ…D¿ÀÂËœ1ÒÓÔÕ×ÿˆð”Þ ”þð¢ ð€¬ “ ð6€ÀÕ¿�ƒ¿Àÿˆðõ " ö ðTŸ¨10¡" 2gþª  ð¢ ð�¬ “ ð6€4Õ¿�ƒ¿Àÿˆð "ö ðTŸ¨11¡" 2gþª  ð¢ ð‚¬ “ ð6€(#Õ¿�ƒ¿Àÿˆðg"wö ðTŸ¨12¡" 2gþª  ðÁ¢ ðƒ¬ “ ð6€@'Õ¿�ƒ¿ÀÿˆðP ¢  v ðSŸ¨4¡" 2gþª  ðÁ¢ ð„¬ “ ð6€ä*Õ¿�ƒ¿Àÿˆð®¢ dv ðSŸ¨5¡" 2gþª  ðÁ¢ ð…¬ “ ð6€üïÓ¿�ƒ¿Àÿˆðg¢ v ðSŸ¨6¡" 2gþª  ðÁ¢ ð†¬ “ 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=ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*‘¬ÿÿÿÿÿÿÿÿ%ñ(ñ+8+0+‘¬0 +î®2ï !€ 0 Å1ð½1 ðLq¸ðU1ð( ð ð¸ðŠ ð¸ “ ð6€œnÕ‡�¿Àÿ ¬ˆð,ù Sðà  Õ ð žð'¢ ð¸ ó ðZ€ÔoÕ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      ð„ 𸠃 ð0€psÕ�¿Àÿ ¬ˆð–:·ðà Õ ð žðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð«  mðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð�«  èðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð «  c ðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðƒ «  Þ ðŠ2 ð ¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðþ«  YðŠ2 ð ¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð dmðŠ2 ð ¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð� dèðŠ2 ð ¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð dc ðŠ2 ð ¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðƒ dÞ ðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðþ dYðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðfÁmðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð�fÁèðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð fÁc ðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðƒ fÁÞ ðŠ2 ð¸ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðþfÁYð… ðF ð䘇 ð¸ ðˆðäqì‡ð„B ð¸ ã ðT…D¿ÀÿÂË8cÒÓÔÕ×ÿðš`ð„B ð¸ ã ðT…D¿ÀÿÂË8cÒÓÔÕ×ÿðÛ ð„B ð¸ ã ðT…D¿ÀÿÂË8cÒÓÔÕ×ÿð� V ð„B ð¸ ã ðT…D¿ÀÿÂË8cÒÓÔÕ×ÿð Ñð 𸠣 ðÒ…BkCì DEÁFÁ�‚ƒ„†Á¿ÀÿÂÄË8cÒÓÔÕ×ÿˆðÿkµ§Oö�µDkì @­ ­ ¬€"ñ¿`ð¥@µ,𻢠𸠃 ð0€¸}Õ¿�ƒ¿ÀÿðâP˜$ ðSŸ¨4¡" 2gÿþª  𻢠𸠃 ð0€È~Õ¿�ƒ¿Àÿðê Ó¾  ðSŸ¨3¡" 2gÿþª  𻢠𸠃 ð0€üÕ¿�ƒ¿Àÿð⢠˜v ðSŸ¨2¡" 2gÿþª  𻢠𸠃 ð0€È‚Õ¿�ƒ¿Àÿðâ' ˜û  ðSŸ¨1¡" 2gÿþª  𻢠𸠃 ð0€P†Õ¿�ƒ¿Àÿð⬘€  ðSŸ¨5¡" 2gÿþª  ð˜2 ð¸ ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðµäkšð˜2 ð ¸ ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðµ_kð˜2 ð!¸ ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðµÚ k� ð˜2 ð"¸ ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðµV k ð˜2 ð#¸ ó ðZ…‡�‚ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðµÑk‡ð|B ð'¸ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðmØ 6�ð|B ð(¸ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðm6”�ð|B ð,¸€ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðmØ ”�ð|B ð0¸ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðèØ 6 ð|B ð1¸ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðè6” ð|B ð5¸€ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðèØ ” ð|B ð9¸ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðc Ø 6ƒ ð|B ð:¸ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðc 6”ƒ ð|B ð>¸€ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðc Ø ”ƒ ð|B ðB¸ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÞ Ø 6þð|B ðC¸ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÞ 6”þð|B ðG¸€ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÞ Ø ”þðô¢ ðQ¸ ó ðZ€À�Õ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð1t  ðjŸ¨Bipartite difference = 1¡" 2gÿþª  ð‚B ðR¸ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿð�= ¨ �ðô¢ ðS¸ ó ðZ€´“Õ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðt S  ðjŸ¨Bipartite difference = 1¡" 2gÿþª  ð‚B ðT¸ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿðÛ= ¨ Ûðô¢ ðU¸ ó ðZ€ì—Õ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðú t Î  ðjŸ¨Bipartite difference = 1¡" 2gÿþª  ð‚B ðV¸ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿðV = ¨ V ðô¢ ðW¸ ó ðZ€¸›Õ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð¢ t v ðjŸ¨Bipartite difference = 1¡" 2gÿþª  ð‚B ðX¸ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿðþ = ¨ þ ðô¢ ðY¸ ó ðZ€Ü Õ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð•t i ðjŸ¨Bipartite difference = 1¡" 2gÿþª  ð‚B ðZ¸ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿðñ= ¨ ñð|B ð\¸€ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðmØ 6þð|B ð]¸€ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðm6”þðè ð^¸ s ðÀ…B¼C‘ DEÁFÁ�ƒ†Á¿ÀÂÅÁËœ1ÒÓÔÕ×ÿˆðÿò¯ä^ö"Žo¼‘ @­ ­ ¬€ðmØ ”þð³¢ ð_¸ ƒ ð0€Ô¥Õ¿�ƒ¿ÀÿðŸ 2 è s  ðSŸ¨3¡" 2gþª  ð³¢ ð`¸ ƒ ð0€¨©Õ¿�ƒ¿Àÿð' (Þû  ðSŸ¨1¡" 2gþª  ð³¢ ða¸ ƒ ð0€À¬Õ¿�ƒ¿Àÿðú S Î  ðSŸ¨2¡" 2gþª  ð³¢ ðb¸ ƒ ð0€L±Õ¿�ƒ¿ÀÿðH Ë ðSŸ¨6¡" 2gþª  ð³¢ ðc¸ ƒ ð0€ð´Õ¿�ƒ¿ÀÿðÐ ¡W¤ ðSŸ¨4¡" 2gþª  ð³¢ ðd¸ ƒ ð0€”¸Õ¿�ƒ¿Àÿðu T I ðSŸ¨5¡" 2gþª  ð´¢ ðe¸ ƒ ð0€8¼Õ¿�ƒ¿ÀÿðÈé ùœ ðTŸ¨12¡" 2gþª  ð´¢ ðf¸ ƒ ð0€Ü¿Õ¿�ƒ¿ÀÿðŒt…` ðTŸ¨10¡" 2gþª  ð´¢ ðg¸ ƒ ð0€€ÃÕ¿�ƒ¿Àÿð#7F÷ ðTŸ¨11¡" 2gþª  ð´¢ ðk¸ ƒ ð0€$ÇÕ¿�ƒ¿Àÿðp` pD  ðTŸ¨15¡" 2gþª  ð´¢ ðl¸ ƒ ð0€ÈÊÕ¿�ƒ¿Àÿð4 ƒ”  ðTŸ¨13¡" 2gþª  ð´¢ ðm¸ ƒ ð0€lÎÕ¿�ƒ¿ÀÿðË®½Ÿ  ðTŸ¨14¡" 2gþª  ð³¢ ðn¸ ƒ ð0€ÒÕ¿�ƒ¿Àÿð Cùë  ðSŸ¨9¡" 2gþª  ð³¢ ðo¸ ƒ ð0€´ÕÕ¿�ƒ¿Àÿð­ s)�  ðSŸ¨7¡" 2gþª  ð³¢ ðp¸ ƒ ð0€XÙÕ¿�ƒ¿Àÿð S ë  ðSŸ¨8¡" 2gþª  ðô¢ ðq¸ ó ðZ€üÜÕ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð†®t¦ ðjŸ¨C(2)¡6 2emª  ðH 𸠃 ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆ‘Љº___PPT10‹ië.ÎË0%u+Dñ='ñ ˜ÿÿÿÿ=ñ @Bñ +î†3ï !€ 0 �2ð•20ðLm¼ð-2ð( ð ð¼ðŠ ð¼ “ ð6€�êÕ‡�¿Àÿ ¬ˆð,ù Sðà  Õ ð žð'¢ ð¼ ó ðZ€ÜëÕ¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM ð�Ÿ¨!Multicolored Subgraph Parallelism¡$" 2"€g€ª4      ð„ ð¼ ƒ ð0€¼ðÕ�¿Àÿ ¬ˆð–:·ðà Õ ð žðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð«  mðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð�«  èðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð «  c ðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðƒ «  Þ ðŠ2 ð ¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðþ«  YðŠ2 ð ¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð dmðŠ2 ð ¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð� dèðŠ2 ð ¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð dc ðŠ2 ð ¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðƒ dÞ ðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðþ dYðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðfÁmðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð�fÁèðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ð fÁc ðŠ2 ð¼ ã ðT…‡�ƒ¿ÂËœ1ÒÓÔÕ×ÿ"ñ¿`ðƒ fÁÞ ðŠ2 ð¼ 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ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÞ Ø ”þðè ðN¼ s ðÀ…BãC‘ DEÁFÁ�ƒ†Á¿ÀÂÅÁËœ1ÒÓÔÕ×ÿˆðÿãq˜1Éaqùã‘ @­ ­ ¬€ðmõ Ø þðè ðO¼ s ðÀ…BãC‘ DEÁFÁ�ƒ†Á¿ÀÂÅÁËœ1ÒÓÔÕ×ÿˆðÿãq˜1Éaqùã‘ @­ ­ ¬€ðmS6þðè ðP¼ s ðÀ…BãC‘ DEÁFÁ�ƒ†Á¿ÀÂÅÁËœ1ÒÓÔÕ×ÿˆðÿãq˜1Éaqùã‘ @­ ­ ¬€ðm±”þðô¢ ðQ¼ ó ðZ€ˆ×¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð1t  ðjŸ¨Bipartite difference = 2¡" 2gÿþª  ð‚B ðR¼ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿð�= ¨ �ðô¢ ðS¼ ó ðZ€T׿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðt S  ðjŸ¨Bipartite difference = 2¡" 2gÿþª  ð‚B ðT¼ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿðÛ= ¨ Ûðô¢ ðU¼ ó ðZ€×¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðú t Î  ðjŸ¨Bipartite difference = 2¡" 2gÿþª  ð‚B ðV¼ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿðV = ¨ V ðô¢ ðW¼ ó ðZ€@׿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð¢ t v ðjŸ¨Bipartite difference = 2¡" 2gÿþª  ð‚B ðX¼ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿðþ = ¨ þ ðô¢ ðY¼ ó ðZ€ "׿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð•t i ðjŸ¨Bipartite difference = 0¡" 2gÿþª  ð‚B ðZ¼ ó ðZ…D¿ÀÿÂË8cÑÒÓÔÕ×ÿðñ= ¨ ñð³¢ ð[¼ ƒ ð0€p'׿�ƒ¿Àÿð“ º pg ðSŸ¨6¡" 2gþª  ð³¢ ð\¼ ƒ ð0€�*׿�ƒ¿Àÿðu 6ìI ðSŸ¨5¡" 2gþª  ð³¢ ð]¼ ƒ ð0€<.׿�ƒ¿Àÿðu û±I ðSŸ¨4¡" 2gþª  ð³¢ ð^¼ ƒ ð0€È2׿�ƒ¿Àÿð » qï  ðSŸ¨3¡" 2gþª  ð³¢ ð_¼ ƒ ð0€l6׿�ƒ¿Àÿðý 7íÑ  ðSŸ¨2¡" 2gþª  ð³¢ ð`¼ ƒ 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Use u, u1 to construct 2nd tree.¡¦*x ee‚ee‚mçÿemeª * ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆãŠÛº___PPT10‹»ë.ÁË`;Lp+¤ÂA�Dñ'ñ  ãÿÿÿÿ=ñ @Bñ DñÚ'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ¶'ñ =ñ%ñ(ñÿÿÿÿDñ^'ñ =ñ%ñ(ñDñ'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñ)ñ?)ñ?DñZ'ñ Ð=ñ.ñ"7ñ ˜ÈBÈBÈBÈBBñ[M -4.44444E-6 2.96296E-6 L 0.11025 -0.12593 *ñ�3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ0Bñ BñAA<ñû*ÿÿÿÿÿÿÿÿDñ'ñ =ñ%ñ(ñÿÿÿÿDñº'ñ =ñ%ñ(ñDñ-'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*3ÿÿÿÿÿÿÿÿ%ñ(ñDñ¡'ñ Ð=ñ-ñi6ñBñ wedge*ñ<3ñ<ñû*3ÿÿÿÿÿÿÿÿDñ-'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñDñ¡'ñ Ð=ñ-ñi6ñBñ wedge*ñ<3ñ<ñû*<ÿÿÿÿÿÿÿÿDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*>ÿÿÿÿÿÿÿÿ%ñ(ñDñß'ñ =ñ%ñ(ñÿÿÿÿDñ‡'ñ =ñ%ñ(ñDñ/'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñ¥'ñ ô=ñ-ñm6ñBñbox(in)*ñ<3ñ<ñû*<ÿÿÿÿÿÿÿÿDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû*<ÿÿÿÿÿÿÿÿ%ñ(ñóDñA'ñ =ñ%ñ(ñÿÿÿÿDñé'ñ =ñ%ñ(ñDñ‘'ñ =ñA@Bñ�Bñ Bñ°Bñ0Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*?ÿÿÿÿÿÿÿÿ%ñ(ñDñÀ'ñ =ñ%ñ(ñÿÿÿÿDñh'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*9ÿÿÿÿÿÿÿÿ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*8ÿÿÿÿÿÿÿÿ%ñ(ñ+p+0+>0 ++0+?0 +îî#ï !€ 0 Ð ðÈ Pð1 ñðP ð( ð ðð� ð £ ð<€à7ã‡�¿Àÿ ¬ˆð,ùYSðà  ã ð žð¢ ð ó ðZ€9ã¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM 𒟨"Existence of Multicolored Subgraph¡$# 2#€g€ª(   ðf2 ð ƒ ð0…‡�ÿƒ¿Àÿ"ñ¿`ð0 dÄ� ðf2 ð ƒ ð0…‡�ÿƒ¿Àÿ"ñ¿`ð0 ¤ � ðf2 ð ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ðÀ tÔ ðf2 ð ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ðÀ d ðf2 ð  ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ðÀ D¤ ðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ðÀ $„ ðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ðÀ d Ä ðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ðÀ ´  ð΢ ð  ƒ ð0€�@ã¿�ƒ¿Àÿð@T:  ðnŸ¨x0¡< 2‚g‚€o€çÿª  ðä¢ ð ƒ ð0€ØDã¿�ƒ¿Àÿð@t± :  ð„Ÿ¨x1¡R 2‚g‚€o€çÿ€g€ª  ðf2 ð ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ðÛ;|ðf2 ð ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð k|ðf2 ð ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ð[»|ðf2 ð ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð›û|ðf2 ð ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð{Û|ðf2 ð ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ð»  |ðf2 ð ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð k |ðf2 ð  ƒ ð0…‡�ÿƒ¿Àÿ"ñ¿`ðÀ ´ ðf2 ð- ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ðÀ ”ô ðf2 ð. ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ðY´¹ðÞðl ðæ C# –  ð0ðCæ # – ð,$ñ@ 0ðì¢ ð ƒ ð0€0Mã¿�ƒ¿Àÿðæ C# =  ð„Ÿ¨x2¡R 2‚g‚€o€çÿ€g€ª  ðn2 ð/ ƒ ð0…‡�ÿƒ¿Àÿÿ"ñ¿`ð 6 s – ðH ð ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÝÿÝC7"IDq­Ii+zŽ2ßãOˆÆŠ¾º___PPT10‹žë.ÁË`;Lp+< ÆDñr'ñ ÿÿÿÿ=ñ @Bñ Dñ-'ñ •ÿÿÿÿ=ñ @BñAñ?%ñ,(ñ <ñ +O%ñ,(ñ <ñ +Dñ¸'ñ =ñ%ñ(ñÿÿÿÿDñ`'ñ =ñ%ñ(ñDñ'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñ)ñ?)ñ?Dñ\'ñ Ð=ñ.ñ$7ñ ˜ÈBÈBÈBÈBBñ]M -1.94444E-6 -3.33333E-6 L 0.28351 -0.20995 *ñ�3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ0Bñ BñAA<ñû* ÿÿÿÿÿÿÿÿDñ¾'ñ =ñ%ñ(ñÿÿÿÿDñf'ñ =ñ%ñ(ñDñ„'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñø'ñ =ñ1ñ :ñBñvisible*ño3ñ>ñ+Bñ#style.visibility<ñû*0ÿÿÿÿÿÿÿÿ%ñ(ñDñ‚'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñDñö'ñ =ñ1ñž:ñBñhidden*ño3ñ>ñ+Bñ#style.visibility<ñû* ÿÿÿÿÿÿÿÿ%ñ(ñDñÞ 'ñ =ñ%ñ(ñÿÿÿÿDñ† 'ñ =ñ%ñ(ñDñ+'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñ)ñ?)ñ?Dñ'ñ Ð=ñ.ñG7ñ ØÈBÈBÈBÈBBñYM -1.11111E-6 3.33333E-6 L 0.16997 -0.3926 *ñ¶3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ@0Bñ BñAApBñ€BñÞÝ­=�BñåI¾<ñû*ÿÿÿÿÿÿÿÿDñ+'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñ)ñ?)ñ?Dñ'ñ Ð=ñ.ñG7ñ ØÈBÈBÈBÈBBñYM -1.11111E-6 3.33333E-6 L 0.19097 -0.3926 *ñ¶3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ@0Bñ BñAApBñ€Bñ9ŽÃ=�BñåI¾<ñû*-ÿÿÿÿÿÿÿÿDñ+'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñ)ñ?)ñ?Dñ'ñ Ð=ñ.ñG7ñ ØÈBÈBÈBÈBBñYM -4.44444E-6 3.33333E-6 L 0.02987 -0.3507 *ñ¶3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ@0Bñ BñAApBñ€BñJŸt<�Bñ�¬3¾<ñû* ÿÿÿÿÿÿÿÿDñ+'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñ)ñ?)ñ?Dñ'ñ Ð=ñ.ñG7ñ ØÈBÈBÈBÈBBñYM 3.33333E-6 -1.85185E-6 L 0.10659 -0.4625 *ñ¶3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ@0Bñ BñAApBñ€Bñ¥OZ=�BñÍÌl¾<ñû*.ÿÿÿÿÿÿÿÿDñ-'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñ)ñ?)ñ?Dñ�'ñ Ð=ñ.ñI7ñ ØÈBÈBÈBÈBBñ[M 1.38889E-6 -2.22222E-6 L 0.05955 -0.43125 *ñ¶3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ@0Bñ BñAApBñ€Bñ33ó<�Bñ$ë\¾<ñû*ÿÿÿÿÿÿÿÿDñ-'ñ =ñ4@Bñ�Bñ Bñ°Bñ%ñ(ñ)ñ?)ñ?Dñ�'ñ Ð=ñ.ñI7ñ ØÈBÈBÈBÈBBñ[M 1.38889E-6 -2.22222E-6 L 0.01979 -0.47315 *ñ¶3ñ>ñ*Bñ ppt_xBñ ppt_y=ñ@0Bñ BñAApBñ€Bñ"""<�Bñy@r¾<ñû*ÿÿÿÿÿÿÿÿ+î£lï !€ 0 ì4ðä4`ðIK ð|4ð( ð ð ð� ð  £ ð<€°]ã‡�¿Àÿ ¬ˆð,ùYSðà  ã ð žð~ ð  s ð*€X^ã¿ÿ¬ˆðÕù‹ šðà ã ð žð¢ ð  ó ðZ€�_ã¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿð�ùkM 𒟨"Existence of Multicolored Subgraph¡$# 2#€g€ª(   ðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð¹ eðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð¹ E¥ðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð¹ %…ðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð¹ eÅð΢ ð  ƒ ð0€lfã¿�ƒ¿Àÿð9 U3  ðnŸ¨x0¡< 2‚g‚€o€çÿª  ðä¢ ð  ƒ ð0€djã¿�ƒ¿Àÿð9 u²3  ð„Ÿ¨x1¡R 2‚g‚€o€çÿ€g€ª  ðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ðÜ<uðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð\¼uðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ðœüuðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð l uðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð¹ µðf2 ð  ƒ ð0…‡�ƒ¿Àÿ"ñ¿`ð¬ uðf2 ð  ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ðÑkË1ðf2 ð  ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ð†pæðf2 ð  ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ð¤Õ5ðf2 ð  ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ð†M­æðf2 ð  ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ð‚X¸âðf2 ð!  ƒ ð0…‡�²²²ƒ¿Àÿ"ñ¿`ðÌ– ö ,ð–2 ð&   ð`…‡�‚ƒ¿ÀÂËœ1ÎÒÓÔÕÿ"ñ¿`ðI¬ < ð—¢ ð*  ó ðZ€”pã¿�ƒ¿ÀÂËœ1ÒÓÔÕ×ÿðm)m �ð,$ñ 0 ðÙŸ¨Find suitable v1, v2.¡” 2eemçÿeemçÿeª  ð¢ ð+  ƒ ð0€Ôyã¿�ƒ¿Àÿðí ŒÉ çð,$ñ 0 ð„Ÿ¨v1¡R 2‚g‚€o€çÿ€g€ª  ð¢ ð,  ƒ ð0€0ã¿�ƒ¿Àÿðí ÷ 4 çð,$ñ 0 ð„Ÿ¨v2¡R 2‚g‚€o€çÿ€g€ª  ðåðl ðS ÐÞ  ðB ðL Ñ× ð,$ñ@ 0ð„B ð- B ã ðT…D¿ÀÿÂËœ1ÒÓÔÕ×ÿðjS ÅÞ ðÝ¢ ð3  ƒ ð0€X…ã¿�ƒ¿ÀÿÿC"ñ¿ÿ?ð; Ðû  ðUŸ¨1¡$ 2€g€ÿþª  ðÅðl ðj€ @ Þ  ðC ðy kA × ð,$ñ@ 0ð„B ð. B ã ðT…D¿ÀÿÂËœ1ÒÓÔÕ×ÿðj€ @ Þ ð½¢ ð4  ƒ ð0€,‰ã¿�ƒ¿Àÿðò; ² û  ðUŸ¨2¡$ 2€g€ÿþª  ðÁðl ðÅS å Þ  ðD ðL Ææ× ð,$ñ@ 0ð„B ð/  ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿðÅS å Þ ð¹¢ ð5  ƒ ð0€XŽã¿�ƒ¿Àÿð ' • û  ðQŸ¨3¡  2€c€ª  ðÅðl ð@ S \ Þ  ðE ðL A ] × ð,$ñ@ 0ð„B ð0  ã ðT…D¿Àxb>ÂËœ1ÒÓÔÕ×ÿð@ S P Þ ð½¢ ð6  ƒ ð0€H’ã¿�ƒ¿ÀÿðÉ ' \ û  ðUŸ¨4¡$ 2€g€ª  ð¢ ð9  ƒ ð0€@–ã¿�ƒ¿ÀÿðB³ð<ð,$ñ   0 ð„Ÿ¨u1¡R 2‚g‚€o€çÿ€g€ª  ð¢ ð:  ƒ ð0€Ä›ã¿�ƒ¿ÀÿðB.k<ð,$ñ  0 ð„Ÿ¨u2¡R 2‚g‚€o€çÿ€g€ª  ðÅ¢ ð;  ƒ ð0€Ü ã¿�ƒ¿ÀÿC"ñ¿ÿ?ðBN^<ðº$ñ   0ˆ†Š4º___PPT10‹± ŠBº___PPT9‹$¬ ðƒŸ¨v1'¡P 2‚g‚€o€çÿ‚f‚ª  ðÅ¢ ð<  ƒ ð0€ð§ã¿�ƒ¿ÀÿC"ñ¿ÿ?ð× 4 D Ñðº$ñ  0ˆ†Š4º___PPT10‹± ŠBº___PPT9‹$¬ ðƒŸ¨v2'¡P 2‚g‚€o€çÿ‚f‚ª  ðÅðl ðÕ ð I ðF ðÖñBð,$ñ@  0ð„B ð7 B ã ðT…D¿ÀÿÂËœ1ÒÓÔÕ×ÿðÕ å Ið½¢ ð=  ƒ ð0€È¦ã¿�ƒ¿Àÿð0 ‰ ð I ðUŸ¨1¡$ 2€g€ÿþª  ðßðz ðžà ®ƒ  ðI  ðˆðD Q a ð,$ñ@  0ðŠB ð8  ó ðZ…D¿ÀÿÂËœ1ÒÓÔÕ×ÿˆðžV ®V ðâ ð>  “ ð6€±ã¿�ƒ¿ÀÿˆðËà ‹ƒ  ðUŸ¨2¡$ 2€g€ÿþª  ðÁðl ð:Þ jI ðG ð× ;kBð,$ñ@   0ð„B ð1 ‚ ã ðT…D¿ÀÂËœ1ÒÓÔÕ×ÿð:Þ jIð¹¢ ð?  ƒ ð0€àµã¿�ƒ¿ÀÿðÂE U ðQŸ¨3¡  2€c€ª  ðÅðl ðµÞ jI ðH ð× ¶kBð,$ñ@  0ð„B ð2 B ã 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Existence of MTP 2. Existence of MHCP Existence of Multicolored Subgraphs 1. Spanning Trees 2. Unicyclic Spanning Subgraphs Conclusion¡z¼²lx2g)g$g2g gªt  P            óŸ¨ Preliminaries¡a€ª  Ÿ LA (proper) k-edge-coloring of a graph G is a mapping from E(G) into {1,& ,k } such that incident edges of G receive distinct colors. Example: 3-edge-coloring of K4 ¡^†x  gg‚ÿþ gg‚gg‚gg‚gg‚ggg‚oçÿg‚ª § óŸ¨ Preliminaries¡a€ª  Ÿ¨nA subgraph in an edge-colored graph is said to be multicolored if no two edges have the same color. Example:¡tox2g€þ g‚3Ìþ(g€þg€þcª4  d  óŸ¨ Preliminaries¡a€ª  Ÿ RG =A*"B is a balanced bipartite graph. Let A = { a1, a2, & , an } and B = { b1, b2, & , bn }, then the bipartite difference of aibj is defined as j  i (mod n). Example:¡ž xx xg‚þgþg‚þ$g€þg‚þg€þg‚þo€þçÿg‚þg€þg‚þo€þçÿg‚þo‚þçÿg€þg‚þg€þg‚þo€þçÿg‚þo€þçÿg‚þo‚þçÿg‚þ g€þg‚ÿþg€þg‚ÿþo‚ÿþçÿg‚ÿþo‚ÿþçÿg€þg‚ÿþg‚þg€þg‚þg‚ÿþg€þg€þcªNU  &  (  óT=Ÿ¨ Preliminaries¡a€ª  Ÿ 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.¡0xgþg‚3Ìþ gþg‚þgþ g‚þgþg‚þ'gþg‚þ0gþg‚þagþgª  óŸ¨ Preliminaries¡a€ª  Ÿ º Idempotent commutative latin square of order n a" n-total-colored Kn a" n-edge-colored Kn,n¡®^xg gÿþ gÿþgg‚gg€g‚g€g‚o‚çÿg€g‚g€g‚o‚çÿo€çÿo‚çÿg‚ªZ  %     óŸ¨Multicolored 1-factor¡a€ª  Ÿ¨æConjecture (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. ¡ çx g�g gÿ™þgg‚gg‚o‚çÿoçÿo‚çÿ<gg‚gg�gg˜›%þ gg‚gg‚oçÿo‚çÿgg‚+gª@/  J   ` ó Ÿ¨Multicolored Subgraph¡a€ª&    Ÿ üGiven an (n 1)-edge-colored Kn, a multicolored Hamiltonian path exists for n > 4. (!?) How about multicolored spanning trees?¡x gg‚g€gg‚o‚çÿgg gg‚"ggÿþgª&  a ó!Ÿ¨ Known Results¡a€ª  Ÿ ,Brualdi-Hollingsworth s Conjecture (1998) If m>2, then in any proper (2m ÿ1)-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"g�ge˜›%þgg‚gg‚€g€g‚o€çÿo‚çÿ$g€g‚g€g�$g€˜›%þg€g€ÿþ=g€ªf B   d    M óU>Ÿ¨Multicolored Tree Parallelism ¡ve€ÿþ a€e€ÿþa€e€ÿþ a€ª  Ÿª óV?Ÿ¨Multicolored Tree Parallelism ¡ve€ÿþ a€e€ÿþa€e€ÿþ a€ª  Ÿ ’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. ¡ÊÊx g�gg‚oçÿo‚çÿ gg‚gƒggÿþgg‚gg‚oçÿo‚çÿ-gg‚g gÿþgª Ê ó"Ÿ¨ Conjecture¡ a€ª  Ÿ ºConstantine s Weak Conjecture (2002) For any m>2, K2m can be (2m ÿ1)-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 ÿ1)-edge-coloring of K2m, all edges can be partitioned into m multicolored isomorphic spanning trees.¡ ^xg�gg˜›%þ gg‚gg‚oçÿo‚çÿ gg‚€Eg€g‚g€ g€ÿþg€g�g€g€˜›%þg€g‚ g€g€ÿþ g€g‚ € g€g‚o€çÿo‚çÿ$g€g‚g€ g€ÿþg€ªJA   ¶   d  óhGŸ¨,Multicolored Hamiltonian Cycle Parallelism ¡Ä-g€$ÿþ c€$g€$ÿþ g€$g€$ÿþg€$c€$g€$ÿþ c€$ª - Ÿ¨ÒDefinition 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. ¡¸Óx g�gg‚oçÿo‚çÿ g*g‚gƒggÿþgg‚gg‚oçÿo‚çÿoçÿ-gg‚#gª Ó ó#Ÿ¨ Known Results¡a€ª  Ÿ 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.¡©xg�gg˜›%þgg‚gg‚oçÿo‚çÿoçÿgg�gg˜›%þgg‚ ggÿþgg‚oçÿo‚çÿoçÿgª © óYAŸ¨ Conjecture¡ a€ª  Ÿ ÂConstantine 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.¡Ê°x2xg�gg˜›%þggÿþ gg‚c€g€g‚o€çÿo‚çÿo€çÿ$g€g‚g€ g€ÿþ g€g€ÿþ5g€ªZš          " ó$Ÿ¨ Known Results¡a€ª  ófFŸ¨ Known Results¡a€ª  ó Ÿ¨!Multicolored Subgraph Parallelism¡""a€ª&     Ÿ¨1. Existence of MTP¡"g(3þª  ó Ÿ¨MTP¡a€ª  Ÿ HConstatine s Weak Conjecture (2002) For any m>2, K2m can be (2m ÿ1)-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.¡´²’lx ’lSg�gg˜›%þ gg‚gg‚oçÿo‚çÿ gg‚€Eg€g‚g€ g€ÿþg€g€þ g�þg€þg€˜›%þg€þg‚þ g€þg‚þo€þçÿo‚þçÿg€þg‚þg€þª@  4   ã  ó Ÿ¨MTP¡a€ª  Ÿ zTheorem 2.2 (Fu and Lo) For m`"2, K2m admits an MTP. Pf: ¡à> g�g ggg‚gg‚oçÿo‚çÿgª > óŸ¨!Multicolored Subgraph Parallelism¡""a€ª&     Ÿ¨2. Existence of MHCP¡"g(3þª  ó Ÿ¨MHCP¡a€ª  Ÿ¨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. ¡Þ’lx’lg�þgþg‚þoþçÿo‚þçÿoþçÿ gþ*g‚þgƒþgþgÿþgþg‚þgþg‚þoþçÿo‚þçÿoþçÿ-gþg‚þgþg3Ìþgþ g�þgþg˜›%þgþg‚þgþg‚þo‚þçÿgþgþª4     ó Ÿ¨MHCP¡a€ª  Ÿ¨ÄLemma 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.¡"Åx g�þ ggþg‚þ gþg‚þgþg‚þgþg‚þgþg‚þgþg‚þo‚þçÿgþg‚þo‚þçÿoþçÿo‚þçÿoþçÿgþg‚þ#gþc€ªN[  .  4  ó Ÿ¨MHCP¡a€ª  Ÿ¨PTheorem 2.5 (Fu and Lo) For each odd integer v 3, Kv admits an MHCP. Pf: ¡Q e�g egee‚ee‚m‚çÿegegª&6   ó Ÿ¨MHCP¡a€ª  Ÿ¨XExample: K15 Start from a 5-total-colored K5. (Consider K5(3) first.)¡ Yx gg‚oçÿ$gg‚oçÿg oçÿ gg‚oçÿ gª Y óŸ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó(Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó)Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó* Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó+!Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  ó,"Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡`  gg‚oçÿgª  óW@Ÿ¨MHCP¡a€ª  Ÿ¨Example: K15 ¡bx gg‚oçÿgª  ó'Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡P  cc‚kçÿcª  ó-#Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡P  cc‚kçÿcª  ó/%Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡P  cc‚kçÿcª  ó.$Ÿ¨MHCP¡a€ª  Ÿ¨ Example: K15¡P  cc‚kçÿcª  ó0&Ÿ¨#Existence of Multicolored Subgraphs¡$$c€$ª(    Ÿ¨1. Spanning Trees¡"g(3þª  ó1'Ÿ¨MST¡a€ª  Ÿ ÄDefinition Æ 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.¡¾ã‚ e�e‚e€e‚e€e‚e€e‚e€e‚m€çÿm‚çÿe€e‚+e€e‚e€e‚e€e‚m€çÿe€e‚m€çÿe€e‚m€çÿe€e‚m€çÿ e€e‚ÿþe€ÿþe‚ÿþe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe‚ÿþm€ÿþçÿe€ÿþe‚ÿþm€ÿþçÿe‚ÿþm€ÿþçÿ e€e‚m€çÿe€e‚m€çÿe€ª ã ó7*Ÿ¨MST¡a€ª  Ÿ¨Example¡6‚e�e€ª  ó8+Ÿ¨MST¡a€ª  Ÿ¨Example¡6‚e�e€ª  ó6)Ÿ¨MST¡a€ª  Ÿ äTheorem 3.1 (Fu and Lo) Let Æ 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. ¡ð~‚t–‚ e� eegg‚g€g‚g€g‚o€çÿo‚çÿg€g‚o€çÿ+g€ag€g€ÿþg‚ÿþo€ÿþçÿg€ c€ ª ó óZBŸ¨MST¡a€ª  Ÿ¨TNotation Tj(i) : the jth spanning tree which was constructed at round i. ¡HU‚ e�ee‚ÿþm‚ÿþçÿmÿþm‚ÿþmÿþee‚ÿþmÿþ0eeÿþe‚ÿþeg€ c€ ª@     : ó4(Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨@Round 1. Pick any one vertex x1. Let T1(1) = x1-star.¡âAxee‚mçÿee‚mçÿmee‚mçÿeª A ó;-Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨Round 2. Pick x2, u.¡€xee‚mçÿee‚eª  ó<.Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨ Round 2. ¡ x eª  ó=/Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨)Round 2. Use u, u1 to construct 2nd tree.¡¦*x ee‚ee‚mçÿemeª * ó>0Ÿ¨Sketch proof of Thm¡a€ª&   Ÿª ó?1Ÿ¨Sketch proof of Thm¡a€ª&   Ÿª ó@2Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨Round 3. Pick x3, u.¡€xee‚mçÿee‚eª  óB3Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ¨Round 3. Pick x3, u.¡€xee‚mçÿee‚eª  ó:,Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ ZDiscussion: (1) Adjust T1(n-1) ’! T1(n), Å"Å"Å"Å"Å"Å", Tn-1(n-1) ’! Tn-1(n). (2) Define Tn(n) = xn-star  {Å"Å"Å"} +{Å"Å"Å"}. (3) Above works whenever |U| 9n-14. (4) |W| 2m-2n2+3n.¡j®x e gg‚oçÿoo‚of€g€g‚o€çÿo€o‚o€g€o€g€g‚o‚çÿo€çÿo€o‚o€ f€ g€g‚o‚çÿo€çÿo€o‚o€ g€g‚o‚çÿo€o‚o€g€g‚o‚çÿg€e€g€e€" g€  g€ ÿþ g‚ ÿþ g€ ÿþ g‚ ÿþ g€ ÿþ g€  g€ ÿþ g‚ ÿþ g€ ÿþ g‚ ÿþ g€ ÿþ g‚ ÿþ o€ ÿþ g€ ÿþ g‚ ÿþ g‚ ª@Q    S óI5Ÿ¨Sketch proof of Thm¡a€ª&   Ÿ øDiscussion: We have n spanning trees if 2m 2n2 + 6n  14. Fixed 2m, we have at least spanning trees. ¡Î ‚1ÿ0 " dÿ‚ÿ " dÿ>ÿ0 " dÿ‚‚ egg‚ggÿþg‚ÿþgÿþg‚ÿþoÿþgÿþg‚ÿþg€ÿþgÿþ gg‚6ggÿþª } óK6Ÿ¨#Existence of Multicolored Subgraphs¡$$c€$ª(    Ÿ¨2. Unicyclic Spanning Subgraph¡"g(3þª@       óL7Ÿ¨MUSS¡a€ª  Ÿ xCorollary 3.2 (Fu and Lo) LetÆbe an arbitrary (2m 1)-edge-coloring of K2m 1. Then, there exist mutually edge-disjoint multicolored unicyclic spanning subgraphs (MUSS).¡ö‚£–e� eegg€g‚g€g‚o€çÿo‚çÿo€çÿ&g€g€g€ÿþ g€g€ÿþ g€g€ÿþg€g€ÿþg€ c€ ªZ  u         óM8Ÿ¨MUSS¡a€ª  Ÿ¨Proof.¡ xeª  óN9Ÿª Ÿ¨ Conclusion¡"  g,3þª  ókHŸ¨ Known Results¡a€ª  ócEŸ¨ Reference¡ a€ª   Ÿ¨¼[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.¡"½x½gª¨    �  A  Š    1 ólIŸ¨ Reference¡ a€ª   Ÿ¨‹[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. Ryser, Neuere Probleme der Kombinatorik, in: Vorträge über Kombinatorik, Oberwolfach, Mathematisches Forschungsinstitute Oberwolfach, Germany, 24-29. ¡ÒŒxBgg‚oçÿo‚çÿ�gg‚oçÿo‚çÿÃgª"V     Ÿ   <      :                                   óŸ¨Thank you for your attention¡a€ª  Ÿª /ðTóó[ó\êâ"xœíX]L\E>3÷îd—?+Ë–(òh¤Ä ,b,H€_ .ºØÕ]¶ü‰5bZ�ðÁ&¼©MÚjû"&} M|ÑF›ø &6ÁŸ‹ ©º& Pðœ;wö¯ »‹)²†9wÎ�9sæžïÌ9Üø:{é£Ë‡~„zØØLcÈÃÆåƒ CÛØÜܤ!ÛfŠöu‚FÀ vÀ~|‘®°-=†À7'ßP¾²(Ä-¡ïZ~M»y*ûK¦ùTšô©gÁ ΄t†’ 8 µ'ž9ÅØò@êoBû=p÷ѯ$¬?õ¤ïÇ3çal.�çºýxòN†cÚOüô�n¿š€~:ÿ ƒàiÞÝ�óÛÐÛŸ(¾Sôÿ¤dâŸ0°þ­Å ·ÿ™IÇ?i¥>ž9IÆØ�ï¼dâŸâO¢ø¿® ^îÛáq ’d\ÈÌØ?©û~¯“aˆ|â`ã@»þôZ\¨´îÐÿûX¼oƦ�èO&íeý2§§ØM!�Ò‚ÈûŸbåú· óÿX@6P ÈSó�@±  Û!l…ØŠôµë}*¯Ø}*Ò¾ùµt7ÜËD§æg‘70¼*š»Úº�ã÷íq ØÜîFǰëÅ&ïKÎÇËÎaÈ6t»<Îak»sÌÚéõ8 [éòyú¼n5yG‡\Î!B¶ÚÖmµ¿>2ä�+…,bütøÊ7V?¿…ž”vä¨}¦ù¨­Ão1?Õkëø³ø0òùføíþGý3�½ÈYÌMÔÍØý½­þm¤û·[‘±Uô6÷Úýôk3g2ôpÆ‘ÐJ†7¥‚Ï*z¹É¢]€¿)�îQ'˜5‰‘„ ¹:u&ê8LÕáŸ3àêD%Tâs- Ö#"pŽ GsJª´Qñv hzîöü³v'.ìJì1ýYÖØ?áÎ › $C$*sBå¼&%¤Š»Ÿ8Æq¢ÆU³},ƒË÷29–Ò 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