Coloring, list coloring, and fractional coloring in intersections of matroids

TL;DR

This paper explores the relationships between various coloring parameters in hypergraphs formed by intersecting multiple matroids, providing bounds on list and fractional chromatic numbers and connecting them to matroidal polytopes.

Contribution

It establishes new bounds on list and fractional chromatic numbers for hypergraphs from intersecting matroids and links these bounds to polytopes and matroidal properties.

Findings

List chromatic number is at most k times the chromatic number.

Fractional chromatic number bounds are connected to matroid polytopes.

Topological tools are used to derive these bounds.

Abstract

It is known that in matroids the difference between the chromatic number and the fractional chromatic number is smaller than 1, and that the list chromatic number is equal to the chromatic number. We investigate the gap within these pairs of parameters for hypergraphs that are the intersection of a given number k of matroids. We prove that in such hypergraphs the list chromatic number is at most k times the chromatic number and at most 2k-1 times the maximum chromatic number among the k matroids. We study the relationship between three polytopes associated with k-sets of matroids, and connect them to bounds on the fractional chromatic number of the intersection of the members of the k-set. This also connects to bounds on the matroidal matching and covering number of the intersection of the members of the k-set. The tools used are in part topological.