List colouring of two matroids through reduction to partition matroids
In the list coloring problem for two matroids, we are given matroids $M_1=(S,{\cal I}_1)$ and $M_2=(S,{\cal I}_2)$ 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 list coloring 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 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 $c\cdot k$. We consider matroid classes that appear naturally in combinatorial optimization problems, namely 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 $M=(S,{\cal I})$ is a matroid in which $S$ can be partitioned into $k$ independent sets, then there exists a partition matroid $N=(S,{\cal J})$ with ${\cal J}\subseteq{\cal I}$ 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. We extend our results by showing that the existence of a matroid $N$ with $χ(N)\leq 2χ(M)$ implies the existence of a matroid $N'$ with $χ(N')\leq 2χ(M')$ for every truncation $M'$ of $M$.