1. Multicolored Subgraphs in an Edge Colored Graph Hung-Lin Fu Department of Applied Mathematics NCTU, Hsin Chu, Taiwan 30050

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). A 3-edge coloring of 5-cycle

3. Facts on Edge-Colorings Let G be a simple graph with maximum degree  (G). Then, the minimum number of colors needed to properly color G,  (G), is either  (G) or  (G) + 1. (Vizing ’ s Theorem) G is of class one if  (G) =  (G) and class two otherwise. K n is of class one if and only if n is even. K n,n is of class one.

4. Rainbow Subgraph Let G be an edge-colored graph. Then a subgraph whose edges are of distinct colors is called a rainbow subgraph of G. It is also known as a heterochromatic subgraph or a multicolored subgraph. Note that we may consider the edge- coloring of the edge-colored graph which is not a proper edge-coloring. In this talk, all edge-colorings are proper edge-colorings. Therefore, a rainbow star can be found easily.

5. Rainbow 1-factor Theorem (Woolbright and Fu, JCD 1998) In any (2m-1)-edge-colored K 2m where m > 2, there exists a rainbow 1-factor. Conjecture (Fu) In any (2m-1)-edge-colored K 2m, there exist 2m-1 edge-disjoint rainbow 1- factors for integers m which are large enough.

6. Theorem (Hatami and Shor, JCT(A) 2008) In any n-edge-colored K n,n, there exists a rainbow matching of size larger than n – (11.053)(log n) 2. Conjecture (Ryser) In any n-edge-colored K n,n, there exists a rainbow 1-factor if n is odd and there exists a rainbow matching of size n – 1 if n is even.

7. What if we can assign the edge-colorings?

8. Room Squares A Room square of side 2m-1 provides a (2m-1)-edge-coloring of K 2m such that 2m-1 edge- disjoint multicolored 1-factors exist. 35 17 28 46 26 48 15 37 13 57 68 24 47 16 38 25 58 23 14 67 12 78 56 34 36 45 27 18

9. Orthogonal Latin Squares A Latin square of order n corresponds to an n-edge-coloring of K n,n. A Latin square of order n with an orthogonal mate provides n edge-disjoint multicolored 1-factors of K n,n. 1 2 3 1 2 3 2 3 1 3 1 2 3 1 2 2 3 1

10. Multicolored Subgraph Conjecture Given an (n-1)-edge- colored K n for even n > n 0, a multicolored Hamiltonian path exists. (By whom?) Problem (Fu and Woolbright) Find “ a ” longest multicolored path in a  (K n )-edge-colored K n.

11. Brualdi-Hollingsworth ’ s Conjecture If m>2, then in any proper edge coloring of K 2m with 2m-1 colors, all edges can be partitioned into m multicolored spanning trees.

12. Multicolored Tree Parallelism K 2m admits a multicolored tree parallelism (MTP) if there exists a proper (2m-1)-edge-coloring of K 2m for which all edges can be partitioned into m isomorphic multicolored spanning trees.

13. K 6 admits an MTP T 1 T 2 T 3 Color 1: 35 46 12 Color 2: 24 15 36 Color 3: 25 34 16 Color 4: 26 13 45 Color 5: 14 23 56 2 54 6 1 3 T1T1

14. Two Conjectures on MTP Constantine ’ s Weak Conjecture For any natural number m > 2, there exists a (2m-1)-edge- coloring of K 2m for which K 2m can be decomposed into m multicolored isomorphic spanning trees.

15. Constantine ’ s Strong Conjecture If m > 2, then in any proper edge coloring of K 2m with 2m-1 colors, all edges can be partitioned into m isomorphic multicolored spanning trees. (Three, so far!)

16. Theorem (Akbari, Alipour, Fu and Lo, SIAM DM) For m is an odd positive integer, then K 2m admits an MTP. Fact. Constantine ’ s Weak Conjecture is true.

17. Edge-colored K n, n is Odd It is well known that K n is of class 2 when n is odd, i. e. the chromatic index of K n is n. In order to find multicolored parallelism, each subgraph has to be of size n. The best candidate is therefore the Hamiltonian cycles of K n. (Unicyclic spanning subgraphs are also great!)

18. K 2m+1 admits a multicolored Hamiltonian cycle parallelism (MHCP) if there exists a proper (2m+1)-edge-coloring of K 2m+1 for which all edges can be partitioned into m multicolored Hamiltonian cycles. MHCP

19. Theorem (Constantine, SIAM DM) If n is an odd prime, then K n admits an MHCP. Conjecture K n admits an MHCP for each odd integer n. The Existence

20. Lemma Let v be a composite odd integer and n is the smallest prime which is a factor of v, say v = mn. If K m admits an MHCP, then K m(n) admits an MHCP. Theorem K v admits an MHCP for each odd integer v. MHCP (Fu and Lo, DM 2009)

21. Let μ be an arbitrary (2m-1)-edge- coloring of K 2m. Then there exist three isomorphic multicolored spanning trees in K 2m for m > 2. Joint work with Y.H. Lo. Can we do it if the edge- coloring is given?

22. Observation If the edge-coloring is arbitrarily given, then finding MTP is going to be very difficult. If we drop “ isomorphism ” requirement for the above case, then may be we can find many multicolored spanning trees of K 2m ?

23. Problem: How many multicolored spanning trees of an edge-colored K 2m can we find if m is getting larger? Guess? Of course, the best result is m.

24. Joint work with Y.H. Lo Theorem For any proper (2m-1)-edge- coloring of K 2m, there exist around m 1/2 mutually edge-disjoint multicolored spanning trees.

25. Definition φ is a (2m-1)-edge-coloring of K 2m, and φ(xy) = c. Define 1.φ x -1 (c) = y and φ y -1 (c) = x 2. xy = x‹c› = y‹c› x y 1 2 3 4 5 = φ x -1 (4)

26. Assume T is a multicolored spanning tree of K 2m with two leaves x 1, x 2. Let the edges incident to x 1 and x 2 be e 1 and e 2 respectively.. Define T[x 1,x 2 ] = T – {e 1,e 2 } + {x 1 ‹c 2 ›, x 2 ‹c 1 ›}. Definition 3 4 u v 1 2 3 4 5 T u v 1 2 5 3 4 T[u,v]

27. Sketch proof 1. Pick any two vertices x ∞, x 1, let T ∞ and T 1 be the stars with centers x ∞, x 1, respectively. x∞x∞ x1x1 T∞T∞

28. 2. Pick x 2, u, v 1, and let the colors be as follows. x∞x∞ x1x1 v1v1 x2x2 u 12 c1c1 c2c2

29. 2. Pick x 2, u, v 1, and let the colors be as follows.. x∞x∞ x1x1 v1v1 x2x2 u 12 c1c1 c2c2 2 1 u1u1

30. 2. Pick x 2, u, v 1, and let the colors be as follows. Find and. Redefine T 1 = T 1 [x 2,v 1 ] x∞x∞ x1x1 v1v1 x2x2 u 2 1 u1u1 T1T1

31. 2. Pick x 2, u, v 1, and let the colors be as follows. Find and. Define x∞x∞ x1x1 x2x2 u 2 1 u1u1 T2T2 c1c1 c2c2

32. 2. The structure of T ∞ so far. x∞x∞ x1x1 T∞T∞ x2x2

33. 2. The structure of T 1 so far. x∞x∞ x1x1 T1T1 x2x2

34. 2. The structure of T 2 so far. x∞x∞ x1x1 T2T2 x2x2

35. 3. Choose x 3, u, v 1, v 2 and let the colors be as follows. x∞x∞ x1x1 x2x2 v1v1 x3x3 u v2v2 12 c1c1 c2c2 34

36. Redefine T 1 = T 1 [x 3, v 1 ]. x∞x∞ x1x1 T1T1 x2x2 v1v1 x3x3 1 3 u v2v2 u1u1

37. 3. Choose x 3, u, v 1, v 2 and let the colors be as follows. Redefine T 2 = T 2 [x 3, v 2 ]. x∞x∞ x1x1 T2T2 x2x2 v1v1 x3x3 u v2v2 24 u2u2

38. 3. Choose x 3, u, v 1, v 2 and let the colors be as follows. Define x∞x∞ x1x1 T3T3 x2x2 x3x3 3 4 u1u1 u2u2

39. Adjust T 1, T 2, T 3 to add an extra T 4. ……… ………until the n-th tree is found.

40. Property: 1. T i and T j are edge-disjoint for i ≠ j ≠ ∞. 2. In each T i, i ≠ ∞, x ∞ is a leaf which is of distant 1 from its center x i. 3. The number of trees n is determined by way of m. T 1, T 2, T 3, …, T n are the desired trees.

41. Corollary For any proper (2m-1)-edge-coloring of K 2m-1, there exist around m 1/2 mutually edge-disjoin multicolored unicyclic spanning subgraphs.

42. 1.Adding an extra vertex, named x ∞, to form a (2m-1)-edge-colored K 2m. 2.Apply Theorem 1 to construct T ∞, T 1, T 2, … 3.Drop x ∞. 4.Adding one specific edge colored the missing color φ (x ∞ x i ) to each T i, for i =1,2,..., Sketch proof

43. Don ’ t Stop!