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

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

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.

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.

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.

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.

What if we can assign the edge-colorings?

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

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

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.

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.

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.

K 6 admits an MTP T 1 T 2 T 3 Color 1: Color 2: Color 3: Color 4: Color 5: T1T1

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.

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!)

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.

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!)

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

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

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)

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?

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 ?

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.

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.

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 = φ x -1 (4)

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 T u v T[u,v]

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∞

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

Don ’ t Stop!