1. 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
2nd JTCCA at Nagoya University

2. Preliminaries
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.
In this talk, all colorings are proper.
A proper 3-edge-coloring of K4

3. Multicolored (Rainbow) subgraph
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. 1 2 3 4 5 1 2 3 4 5
A multicolored 5-cycle

4. Rainbow 1-factor
Conjecture (Transversal) In any n-edge-colored Kn,n, there exists a rainbow 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 rainbow 1-factor.
Open problem
Can you find two edge-disjoint rainbow 1-factors in any (2m-1)-edge-colored K2m where m > 2?

5. Rainbow Hamiltonian Path
Given an (n–1)-edge-colored Kn, a rainbow Hamiltonian path exists for n > n0. (!?) (I conjecture that this is a true.)
How about rainbow spanning trees?
We shall use multicolored spanning trees in the following slides.

6. Known Results
Brualdi-Hollingsworth’s Conjecture (1998, JCT(B)) If m > 2, then in any proper (2m－1)-edge-coloring of K2m, all edges can be partitioned into m multicolored spanning trees. (They found two.)
Theorem (Krussel, Marshall and Verall, 2000, Ars Combin.) Three edge-disjoint multicolored spanning trees always exist.

7. Multicolored Tree Parallelism
Introduction
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 x5x6 x1 x2 x3 x4 x5 x6 T1

8. 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.

9. Conjectures
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 isomorphic multicolored 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 isomorphic multicolored spanning trees.

10. Continued
Constantine’s Strong Conjecture on odd order (2005) In any proper (2m+1)-edge-coloring of K2m+1, all edges can be partitioned into m multicolored isomorphic spanning unicyclic subgraphs.
Unicyclic: a graph with exactly one cycle.

11. Multicolored Hamiltonian Cycle Parallelism
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.

12. Existence of multicolored subgraphs
First, we consider the existence of multicolored spanning trees and the first one is easy to find. (See it?)
In order to find more edge-disjoint multicolored spanning trees, we apply the following idea: Edge-switching.

13. MST x e2 e1 y1 z2 y2 z1

14. MST
φ 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. (Obtained by switching e1 and e2 with y1z1 and y2z2 respectively.)

15. MST T[x;y1,y2;z1,z2] x e2 e1 y1 z2 y2 Fact:
If φ(e1) = φ(y2z2) and φ(e2) = φ(y1z1),
then T[x;y1,y2;z1,z2] is multicolored. z1

16. MST
Tj(i) : the j –th spanning tree which was constructed at round i.

17. Idea of finding more MST’s
Round 1. Pick any one vertex x Let T1(1) = x1-star. x0 x1

18. Round 2. Pick x2, u. Reserve x2 as the root of the second MST.
Find a suitable v1. 1 2 x2 u 2 v1 1
v1' u1

19. Keep going! Round 2. Let T1(2)=T1(1)[x1;x2,v1;u1,v1']. x0 x1 u x2 v1

20. Round 2. Use u, u1 to construct 2nd MST.x0 x1 2 u' c u x2 v1 c
u1' 2
v1' u1

21. Round 2. Use u, u1 to construct 2nd tree.
Let T2(2)= x2-star
– x2u – x2u1
+uu' +u1u1' . x0 x1 u' 2 u x2 v1 c
u1' v' u1

22. Which vertex is our next root?

23. Round 3. Pick x3, u. Find suitable v1, v2. Let :
T1(3)=T1(2)[x1;x3,v1;u1,v1']
T2(3)=T2(2)[x2;x3,v2;u2,v2'] x0 x1 x2
T3(3)= x3-star
– x3u – x3u1– x3u1’ 1 2
+uu' +u1u1' +u2u2' u' 4 c c 3 x3 u 4 v1 v2 c
u1' u1 u2 3
u2'

24. Discussion: (1) Adjust T1(n-1) → T1(n), ⋅⋅⋅⋅⋅⋅, Tn-1(n-1) → Tn-1(n).
(2) Define Tn(n) = xn- star – {⋅⋅⋅} +{⋅⋅⋅}.
(3) The above process works whenever |U| 9n -14.
(4) |W| 2m - 2n2 + 3n. W x0 x1
xn- xn U

25. MST
Theorem 3.1
Let φ 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.

26. Existence of Multicolored Subgraphs
2. Unicyclic Spanning Subgraph

27. MUSS
Existence of Multicolored Subgraph
Corollary 3.2
Letφbe an arbitrary (2m–1)-edge-coloring of K2m–1. Then, there exist mutually edge-disjoint multicolored unicyclic spanning subgraphs (MUSS).

28. MUSS Proof. (2m–1)-edge-colored K2m-1 T1 → C1=T1-x0+e1 T2 →
Existence of Multicolored Subgraph
Proof.
(2m–1)-edge-colored K2m-1 T1 →
C1=T1-x0+e1 T2 →
C2=T2-x0+e2 x1 x0
. . . x2
(2m–1)-edge-colored K2m

29. Conclusion K2m m m Assigned Given K2m+1 3 trees 2 coloring if m>13
(Isomorphic)
Given
K2m
(spanning tree)
3 trees
if m>13
K2m+1
(spanning unicyclic subgraph) 2 m m

30. New result
I just heard from Prof. Chris Rodger that Paul Horn proved the following result: In a (2m-1)-edge-colored complete graph of order 2m, there exist m multicolored spanning trees. The proof uses a probabilistic method.
By the way, the  is quite small.

31. 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,
[2] R. A6. Brualdi and S. Hollingsworth, Multicolored trees in complete graphs, J. Combin. Theory Ser. B 68 (1996), No. 2,
[3] G. M. Constantine, Multicolored parallelisms of isomorphic spanning trees, Discrete Math. Theor. Compu. Sci. 5 (2002), No. 1,
[4] G. M. Constantine, Edge-disjoint isomorphic multicolored trees and cycles in complete graphs, SIAM J. Discrete Math. 18 (2005), No. 3,
[5] H. L. Fu and Y. H. Lo, Multicolored parallelisms of Hamiltonian cycles, Discrete Math. 309 (2009), No. 14,

32. 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 K2n, J. Combin. Des. 6 (1998), 1-20.
[8] J. Krussel, S. Marshal and H. Verral, Spanning trees orthogonal to one-factorizations of K2n, Ars Combin. 57 (2000),
[9] H. J. Ryser, Neuere Probleme der Kombinatorik, in: Vorträge über Kombinatorik, Oberwolfach, Mathematisches Forschungsinstitute Oberwolfach, Germany,

33. Thank you for your attention!