Coloring the intersection of two matroids

TL;DR

This work removes the divisibility restriction in the prior result on coloring the intersection of two matroids and shows that, for any positive integers $p,q$, the intersection $\mathcal{M}\cap\mathcal{N}$ is at most $\chi(\mathcal{M})+\chi(\mathcal{N})$-colorable and $p+q$-list-colorable. The authors develop a topology-based parameter $\eta$ and a new combinatorial parameter $\nu_{p,q}$ to connect topological connectivity with combinatorial structure, enabling a unified bound on the chromatic and list-chromatic numbers of matroid intersections. The approach hinges on circuit representations via hypergraphs and a series of deletions/contractions analyzed through Mayer–Vietoris-type inequalities, culminating in sharp bounds and tightness demonstrations. The results significantly advance matroid coloring theory by integrating topological methods with matroid intersection techniques and clarifying the limits of coloring for intersections without divisibility constraints.

Abstract

A result [The intersection of a matroid and a simplicial complex, Trans. Amer. Math. Soc. 358] from 2006 of Aharoni and the first author of this paper states that for any two positive integers $p,q$, where $p$ divides $q$, if a matroid $\mathcal{M}$ is $p$-colorable and a matroid $\mathcal{N}$ is $q$-colorable then $\mathcal{M} \cap \mathcal{N}$ is $(p+q)$-colorable. In this paper we show that the assumption that $p$ divides $q$ is in fact redundant, and we also prove that $\mathcal{M} \cap \mathcal{N}$ is even $p+q$ list-colorable. The result uses topology and relies on a new parameter yielding a lower bound for the topological connectivity of the intersection of two matroids.

Theorems & Definitions (25)

• Definition 1.1
• Definition 1.2
• Definition 1.3
• Theorem 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.3
• Theorem 3.1
• Theorem 3.2
• Theorem 3.3