Multicolored Subgraphs in an Edgecolored Complete Graph HungLin
Multicolored Subgraphs in an Edge-colored Complete Graph Hung-Lin Fu (Work jointly with Yuan-Hsun Lo and Chris Rodger) Department of Applied Mathematics National Chiao Tung University Hsin Chu, Taiwan 30010 2 nd JTCCA at Nagoya University
Preliminaries n A (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. n In this talk, all colorings are proper. A proper 3 -edge-coloring of K 4 2
Multicolored (Rainbow) subgraph n A subgraph H in an edge-colored graph G is a rainbow subgraph of G if no two edges in H have the same color. 3 4 A multicolored 5 -cycle 1 5 2 5 4 3 1 2 3
Rainbow 1 -factor n Conjecture (Transversal) In any n-edge-colored Kn, n, there exists a rainbow 1 -factor for each odd integer n. n Theorem (Woolbright and Fu, 1998) In any (2 m-1)-edge-colored K 2 m where m > 2, there exists a rainbow 1 -factor. n Open problem Can you find two edge-disjoint rainbow 1 -factors in any (2 m-1)-edge-colored K 2 m where m > 2? 4
Rainbow Hamiltonian Path n Given an (n– 1)-edge-colored Kn, a rainbow Hamiltonian path exists for n > n 0. (!? ) (I conjecture that this is a true. ) n How about rainbow spanning trees? n We shall use multicolored spanning trees in the following slides. 5
Known Results n Brualdi-Hollingsworth’s Conjecture (1998, JCT(B)) If m > 2, then in any proper (2 m-1)-edge-coloring of K 2 m, all edges can be partitioned into m multicolored spanning trees. (They found two. ) n Theorem (Krussel, Marshall and Verall, 2000, Ars Combin. ) Three edge-disjoint multicolored spanning trees always exist. 6
Multicolored Tree Parallelism T 1 T 2 Introduction T 3 x 2 Color 1 x 3 x 5 x 4 x 6 x 1 x 2 Color 2 x 2 x 4 x 1 x 5 x 3 x 6 Color 3 x 2 x 5 x 3 x 4 x 1 x 6 Color 4 x 2 x 6 x 1 x 3 x 4 x 5 Color 5 x 1 x 4 x 2 x 3 x 5 x 6 x 4 x 5 x 6 x 3 x 1 T 1 7
n K 2 m admits a multicolored tree parallelism (MTP) if there exists a proper (2 m– 1)-edge-coloring of K 2 m for which all edges can be partitioned into m isomorphic multicolored spanning trees. 8
Conjectures n Constantine’s Weak Conjecture (2002) For any m > 2, K 2 m can be (2 m-1)-edge-colored in such a way that the edges can be partitioned into m isomorphic multicolored spanning trees. n Constantine’s Strong Conjecture (2002) If m > 2, then in any proper (2 m-1)-edge-coloring of K 2 m, all edges can be partitioned into m isomorphic multicolored spanning trees. 9
Continued Constantine’s Strong Conjecture on odd order (2005) In any proper (2 m+1)-edge-coloring of K 2 m+1, all edges can be partitioned into m multicolored isomorphic spanning unicyclic subgraphs. Unicyclic: a graph with exactly one cycle. 10
Multicolored Hamiltonian Cycle Parallelism n K 2 m admits a multicolored Hamiltonian cycle parallelism (MHCP) if there exists a proper (2 m+1)edge-coloring of K 2 m+1 for which all edges can be partitioned into m multicolored Hamiltonian cycles. 11
Existence of multicolored subgraphs First, we consider the existence of multicolored spanning trees and the first one is easy to find. (See it? ) n In order to find more edge-disjoint multicolored spanning trees, we apply the following idea: Edge-switching. n 12
MST x e 1 y 1 e 2 z 2 y 2 z 1 13
MST φ is a (2 m– 1)-edge-coloring of K 2 m, and T is a multicolored spanning tree with root x. If x is incident to two leaves e 1 = xy 1 and e 2 = xy 2, then let T[x; y 1, y 2; z 1, z 2] = T – e 1 – e 2 + y 1 z 1 + y 2 z 2, for some z 1, z 2. (Obtained by switching e 1 and e 2 with y 1 z 1 and y 2 z 2 respectively. ) 14
MST x e 1 y 1 z 1 e 2 z 2 T[x; y 1, y 2; z 1, z 2] y 2 Fact: If φ(e 1) = φ(y 2 z 2) and φ(e 2) = φ(y 1 z 1), then T[x; y 1, y 2; z 1, z 2] is multicolored. 15
MST Tj(i) : the j –th spanning tree which was constructed at round i. 16
Idea of finding more MST’s Round 1. Pick any one vertex x 1. Let T 1(1) = x 1 -star. x 0 x 1 17
Round 2. Pick x 2, u. Reserve x 2 as the root of the second MST. x 0 x 1 1 u x 2 2 u 1 Find a suitable v 1. 2 v 1 1 v 1 ' 18
Keep going! Round 2. Let T 1(2)=T 1(1)[x 1; x 2, v 1; u 1, v 1']. x 0 u x 2 2 u 1 x 1 v 1 ' 19
Round 2. Use u, u 1 to construct 2 nd MST. x 0 u' 2 u c 2 u 1 x 2 c x 1 v 1 u 1' v 1 ' 20
Round 2. Use u, u 1 to construct 2 nd tree. Let T 2(2)= x 2 -star x 0 u' 2 u x 2 +uu' +u 1 u 1'. x 1 v 1 c u 1 – x 2 u 1 u 1' v' 21
Which vertex is our next root? x 0 x 1 x 2 22
Round 3. Pick x 3, u. Find suitable v 1, v 2. Let : T 1(3)=T 1(2)[x 1; x 3, v 1; u 1, v 1'] x 0 x 1 1 u' 3 c u 1' 4 u 1 T 2(3)=T 2(2)[x 2; x 3, v 2; u 2, v 2'] x 2 T 3(3)= x 3 -star – x 3 u 1– x 3 u 1’ +uu' +u 1 u 1' +u 2 u 2' 2 c u u 2 4 x 3 v 1 3 u 2' v 2 23
Discussion: (1) Adjust T 1(n-1) → T 1(n), ⋅⋅⋅⋅⋅⋅, Tn-1(n-1) → Tn-1(n). (2) Define Tn(n) = xn- star – {⋅⋅⋅} +{⋅⋅⋅}. (3) The above process works whenever |U| (4) |W| 9 n -14. 2 m - 2 n 2 + 3 n. W x 0 U x 1 xn- xn 24
MST Theorem 3. 1 Let φ be an arbitrary (2 m– 1)-edge-coloring of K 2 m and x 0 be an arbitrary vertex. Then, there exist mutually edge-disjoint multicolored spanning trees, each of which contains a pendent vertex x 0. 25
Existence of Multicolored Subgraphs 2. Unicyclic Spanning Subgraph 26
MUSS Existence of Multicolored Subgraph Corollary 3. 2 Letφbe an arbitrary (2 m– 1)-edge-coloring of K 2 m– 1. Then, there exist mutually edge-disjoint multicolored unicyclic spanning subgraphs (MUSS). 27
MUSS Existence of Multicolored Subgraph Proof. (2 m– 1)-edge-colored K 2 m-1 x 1 T 1 → C 1=T 1 -x 0+e 1 T 2 → C 2=T 2 -x 0+e 2 x 2 . . . x 0 (2 m– 1)-edge-colored K 2 m 28
Conclusion coloring Assigned (Isomorphic) K 2 m m K 2 m+1 m (spanning tree) (spanning unicyclic subgraph) Given (Isomorphic) 3 trees if m>13 2 29
New result I just heard from Prof. Chris Rodger that Paul Horn proved the following result: In a (2 m-1)-edge-colored complete graph of order 2 m, there exist m multicolored spanning trees. The proof uses a probabilistic method. n By the way, the is quite small. n
Reference [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. A 6. 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. 31
Continued … [6] H. L. Fu and Y. H. Lo, Multicolored isomorphic spanning trees in complete graphs, Ars Combinatoria, to appear. [7] H. L. Fu and D. E. Woolbright, On the exists of rainbows in 1 factorizations of K 2 n, J. Combin. Des. 6 (1998), 1 -20. [8] J. Krussel, S. Marshal and H. Verral, Spanning trees orthogonal to onefactorizations of K 2 n, Ars Combin. 57 (2000), 77 -82. [9] H. J. Ryser, Neuere Probleme der Kombinatorik, in: Vorträge über Kombinatorik, Oberwolfach, Mathematisches Forschungsinstitute Oberwolfach, Germany, 24 -29. 32
Thank you for your attention! 33
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