List Coloring of Two Matroids through Reduction to Partition Matroids

In the list coloring problem for two matroids, we are given matroids $M_1=(S,{\mathcal{I}}_1)$ and $M_2=(S,{\mathcal{I}}_2)$ on the same ground set $S$, and the goal is to determine the smallest number $k$ such that, given arbitrary lists $L_s$ of $k$ colors for $s\in S$, it is possible to choose a color 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 coloring theorem for bipartite graphs gives the answer. However, not much is known about the general case. One of the main open questions is to decide if there exists a constant $c$ such that if the coloring number is $k$ (i.e., the ground set can be partitioned into $k$ common independent sets), then the list coloring number is at most $c\cdot k$. In the present paper, we consider matroid classes that appear naturally in combinatorial and graph optimization problems, specifically graphic matroids, paving matroids and gammoids. We show that if both matroids are from these fundamental classes, then the list coloring number is at most twice the coloring number. The proof is based on a new approach that reduces a matroid to a partition matroid without increasing its coloring number too much and might be of independent combinatorial interest. In particular, we show that if $M=(S,{\mathcal{I}})$ is a matroid in which $S$ can be partitioned into $k$ independent sets, then there exists a partition matroid $N=(S,{\mathcal{J}})$ with ${\mathcal{J}}\subseteq{\mathcal{I}}$ in which $S$ can be partitioned into (A) $k$ independent sets if $M$ is a transversal matroid, (B) $2k-1$ independent sets if $M$ is a graphic matroid, (C) $\lceil kr/(r-1)\rceil$ independent sets if $M$ is a paving matroid of rank $r$, and (D) $2k-2$ independent sets if $M$ is a gammoid. It should be emphasized that in cases (A), (B), and (D) the rank of $N$ is the same as that of $M$. We further extend our results to a much broader family by showing that taking direct sum, homomorphic image, or truncation of matroids from these classes results in a matroid admitting a reduction to a partition matroid with coloring number at most twice the original one.