On the choosability of $H$ -minor-free graphs.

Given a graph $H$ , let us denote by $f_\chi (H)$ and $f_\ell (H)$ , respectively, the maximum chromatic number and the maximum list chromatic number of $H$ -minor-free graphs. Hadwiger's famous colouring conjecture from 1943 states that $f_\chi (K_t)=t-1$ for every $t \ge 2$. A closely related problem that has received significant attention in the past concerns $f_\ell (K_t)$ , for which it is known that $2t-o(t) \le f_\ell (K_t) \le O(t (\!\log \log t)^6)$. Thus, $f_\ell (K_t)$ is bounded away from the conjectured value $t-1$ for $f_\chi (K_t)$ by at least a constant factor. The so-called $H$ -Hadwiger's conjecture, proposed by Seymour, asks to prove that $f_\chi (H)={\textrm{v}}(H)-1$ for a given graph $H$ (which would be implied by Hadwiger's conjecture). In this paper, we prove several new lower bounds on $f_\ell (H)$ , thus exploring the limits of a list colouring extension of $H$ -Hadwiger's conjecture. Our main results are: For every $\varepsilon \gt 0$ and all sufficiently large graphs $H$ we have $f_\ell (H)\ge (1-\varepsilon)({\textrm{v}}(H)+\kappa (H))$ , where $\kappa (H)$ denotes the vertex-connectivity of $H$. For every $\varepsilon \gt 0$ there exists $C=C(\varepsilon)\gt 0$ such that asymptotically almost every $n$ -vertex graph $H$ with $\left \lceil C n\log n\right \rceil$ edges satisfies $f_\ell (H)\ge (2-\varepsilon)n$. The first result generalizes recent results on complete and complete bipartite graphs and shows that the list chromatic number of $H$ -minor-free graphs is separated from the desired value of $({\textrm{v}}(H)-1)$ by a constant factor for all large graphs $H$ of linear connectivity. The second result tells us that for almost all graphs $H$ with superlogarithmic average degree $f_\ell (H)$ is separated from $({\textrm{v}}(H)-1)$ by a constant factor arbitrarily close to $2$. Conceptually these results indicate that the graphs $H$ for which $f_\ell (H)$ is close to the conjectured value $({\textrm{v}}(H)-1)$ for $f_\chi (H)$ are typically rather sparse. [ABSTRACT FROM AUTHOR]