Improved upper bounds on the list chromatic number of $K_t$-minor-free graphs

It remains open whether every $K_t$-minor-free graph is $O(t)$-choosable. Postle proved that every $K_t$-minor-free graph has choice number $O(t(\log \log t{)}^6)$. At the end of an earlier version of a paper establishing an $O(t\log \log t)$ bound on the chromatic number of $K_t$-minor-free graphs, Delcourt and Postle remarked that their methods, combined with Postle’s earlier techniques, yield an $O(t(\log \log t{)}^2)$ bound on the choice number. In this paper, we first prove that every $n$-vertex $K_t$-minor-free graph has choice number $O(t\log (2+n\mathrm{/}t))$. Using this bound as a key ingredient, we follow the approach outlined by Delcourt and Postle to prove that every $K_t$-minor-free graph is $O(t\log \log t)$-choosable.