Coloring, list coloring, and fractional coloring in intersections of matroids
Ron Aharoni, Eli Berger, He Guo, Dani Kotlar
TL;DR
The paper investigates colorings in intersections of $k$ matroids, focusing on list and fractional colorings within the class $MINT_k$. It develops a topological framework and a polytope viewpoint to bound $\chi_\ell(\mathcal{C})$ in terms of $\chi(\mathcal{C})$ and the $\chi(\mathcal{M}_i)$, obtaining $\chi_\ell(\mathcal{C}) \le k\chi(\mathcal{C})$ for $\mathcal{C}\in MINT_k$ and, for partition matroids, $\chi_\ell(\mathcal{C}) \le k\max_i \chi(\mathcal{M}_i)$; a stronger $(2k-1)$-bound holds for general matroids. The work connects fractional colorings to polytopes $P(\mathcal{C})$, $Q(\mathcal{C})$, and $R(\mathcal{L})$, establishing equivalences and bounding relations that generalize Edmonds’ two-matroid theorem to $k$ matroids, and relates these bounds to matroidal matching/covering numbers via the Mayer–Vietoris framework and topological expansion. It also extends to weighted settings, proving $\chi^*(\mathcal{C},\vec{w}) \le \Delta(\mathcal{C},\vec{w})$ for vertex-weighted complexes and showing equality for matroids, with consequences for weighted Ryser-type conjectures in partition matroid intersections. The final sections synthesize fractional list colorings and fractional list colorability, establishing that $\chi^*(\mathcal{C}) = clr(\mathcal{C}) = chr(\mathcal{C})$, and highlighting the tight interplay between topology, polytopes, and matroid theory in combinatorial coloring problems.
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.