Linear list-colorability of minor-free graphs

Determine whether every $K_t$-minor-free graph is $O(t)$-choosable, equivalently, whether there is an absolute constant $C$ such that every $K_t$-minor-free graph has choice number at most $Ct$.

Background

The paper studies the list chromatic number (choice number) of graphs excluding a complete graph as a minor. The direct list-coloring analogue of Hadwiger’s conjecture is false in its exact $(t-1)$-color form, but prior work established increasingly strong upper bounds, culminating before this paper in an $O(t(clogootnotesize nobreak\log t)^6)$ bound.

The paper improves the upper bound to $O(tclogootnotesize nobreak\log t)$ by first proving an $O(t\log(2+n/t))$ bound for $n$-vertex $K_t$-minor-free graphs. This does not resolve the explicitly stated linear $O(t)$ question, which remains the central conjectural target identified in the paper.