List colouring of two matroids through reduction to partition matroids
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In the list colouring problem for two matroids, we are given matroids M_1=(S,I_1) and M_2=(S,I_2) on the same ground set S, and the goal is to determine the smallest number k such that given arbitrary lists L_s of k colours for s∈ S, it is possible to choose a colour from each list so that every monochromatic set is independent in both M_1 and M_2. When both M_1 and M_2 are partition matroids, Galvin's celebrated list colouring theorem for bipartite graphs gives the answer. One of the main open questions is to decide if there exists a constant c such that if the colouring number is k (i.e., the ground set can be partitioned into k independent sets), then the list colouring number is at most c· k. In the present paper, we consider matroid classes that appear naturally in combinatorial and graph optimization problems, namely graphic matroids, paving matroids and gammoids. We show that if both matroids are from these fundamental classes, then the list colouring number is at most twice the colouring number. The proof is based on a novel approach that reduces a matroid to a partition matroid, and might be of independent combinatorial interest. In particular, we show that if M=(S,I) is a matroid in which S can be partitioned into k independent sets, then there exists a partition matroid N=(S,J) with J⊆I in which S can be partitioned into (A) kr/(r-1) independent sets if M is a paving matroid of rank r, (B) 2k-1 independent sets if M is a graphic matroid, (C) k independent sets if M is a transversal matroid, and (D) 2k-2 independent sets if M is a gammoid. We also show how the reduction technique can be extended to strongly base orderable matroids that might serve as a useful tool in problems related to packing bases of two matroids.
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