The list chromatic number of the intersection of two generalized partition matroids

TL;DR

This paper proves a conjecture that the list chromatic number equals the chromatic number for the intersection of any two generalized partition matroids, extending Galvin's theorem and confirming related conjectures.

Contribution

It establishes that the equality of list chromatic number and chromatic number holds for the intersection of any two generalized partition matroids, confirming conjectures by Kiraly, Berczi, Aharoni, and Berger.

Findings

Proved the conjecture for generalized partition matroids

Extended Galvin's theorem to a broader class of matroids

Confirmed the Aharoni--Berger conjecture

Abstract

A famous theorem of Galvin states that the list chromatic number of the intersection of two partition matroids equals its chromatic number. Kiraly and Berczi et. al. conjectured that this equality holds for any two matroids. We prove this conjecture and a conjecture by Aharoni--Berger for any two generalized partition matroids.