List coloring of matroids and base exchange properties

TL;DR

This paper generalizes Seymour's list coloring theorem for matroids to fixed, non-uniform list sizes and derives new base exchange properties from these list coloring conditions.

Contribution

It extends list coloring results for matroids to non-uniform list sizes and provides explicit conditions for list colorability, leading to new base exchange properties.

Findings

Generalized Seymour's theorem to fixed, non-uniform list sizes

Derived explicit necessary and sufficient conditions for list colorability

Established new base exchange properties from list coloring conditions

Abstract

A coloring of a matroid is an assignment of colors to the elements of its ground set. We restrict to proper colorings - those for which elements of the same color form an independent set. Seymour proved that a $k$-colorable matroid is also colorable from any lists of size $k$. We generalize this theorem to the case when lists have still fixed sizes, but not necessarily equal. For any fixed size of lists assignment $\ell$, we prove that, if a matroid is colorable from a particular lists of size $\ell$, then it is colorable from any lists of size $\ell$. This gives an explicit necessary and sufficient condition for a matroid to be list colorable from any lists of a fixed size. As an application, we show how to use our condition to derive several base exchange properties.