Improved lower bound for the list chromatic number of graphs with no $K_t$ minor

TL;DR

This paper establishes a new lower bound on the list chromatic number of graphs with no $K_t$-minor, showing it is at least approximately 2 times $t$, thus refuting a previous conjecture and advancing understanding of graph coloring limits.

Contribution

It provides the first lower bound of about 2 times $t$ for the list chromatic number of $K_t$-minor-free graphs, improving previous bounds and using probabilistic methods.

Findings

Lower bound of approximately 2t for list chromatic number

Refutes the conjecture that c=1.5 suffices

Uses probabilistic construction for examples

Abstract

Hadwiger's conjecture asserts that every graph without a $K_t$-minor is $(t-1)$-colorable. It is known that the exact version of Hadwiger's conjecture does not extend to list coloring, but it has been conjectured by Kawarabayashi and Mohar (2007) that there exists a constant $c$ such that every graph with no $K_t$-minor has list chromatic number at most $ct$. More specifically, they also conjectured that this holds for $c=\frac{3}{2}$. Refuting the latter conjecture, we show that the maximum list chromatic number of graphs with no $K_t$-minor is at least $(2-o(1))t$, and hence $c \ge 2$ in the above conjecture is necessary. This improves the previous best lower bound by Bar\'{a}t, Joret and Wood (2011), who proved that $c \ge \frac{4}{3}$. Our lower-bound examples are obtained via the probabilistic method.