Disproof of a Conjecture by Woodall

TL;DR

This paper disproves Woodall's conjecture by constructing graphs without certain minors that have higher list chromatic numbers than previously conjectured, showing the conjecture does not hold in general.

Contribution

It provides a strong disproof of Woodall's conjecture by demonstrating the existence of graphs with no $K_{s,t}$-minor yet high list chromatic number for large parameters.

Findings

Counterexamples for large s and t with high list chromatic number

Disproof of Woodall's conjecture in a strong form

Existence of graphs without $K_{s,t}$-minor with chromatic number exceeding previous bounds

Abstract

In 2001, Woodall conjectured that for every pair of integers $s,t \ge 1$, all graphs without a $K_{s,t}$-minor are $(s+t-1)$-choosable. In this note we refute this conjecture in a strong form: We prove that for every choice of constants $\varepsilon>0$ and $C \ge 1$ there exists $N=N(\varepsilon,C) \in \mathbb{N}$ such that for all integers $s,t $ with $N \le s \le t \le Cs$ there exists a graph without a $K_{s,t}$-minor and list chromatic number greater than $(1-\varepsilon)(2s+t)$.