On-line list coloring of matroids

TL;DR

This paper extends Seymour's theorem to the online setting for matroid list coloring, proving that proper colorings are possible under adversarial list revelations and providing a necessary and sufficient condition for matroid list colorability.

Contribution

It generalizes Seymour's theorem to online list coloring of matroids and introduces a weighted version with varying list sizes.

Findings

Online list coloring of matroids is possible with adversarial list revelation.

A weighted version of the theorem is established.

A simple proof of the multiple basis exchange property is provided.

Abstract

A coloring of a matroid is proper if elements of the same color form an independent set. A theorem of Seymour asserts that a k-colorable matroid is also colorable from any lists of size k. In this note we generalize this theorem to the on-line setting. We prove that a coloring of a matroid from lists of size k is possible even if appearances of colors in the lists are recovered color by color by an adversary, while our job is to assign a color immediately after it is recovered. We also prove a more general weighted version of our result with lists of varying sizes. In consequence we get a simple necessary and sufficient condition for matroid list colorability in general case. The main tool we use is the multiple basis exchange property, which we give a simple proof.