Hadwiger’s conjecture asserts that every graph without a $K_t$ -minor is $(t-1)$ -colourable. It is known that the exact version of Hadwiger’s conjecture does not extend to list colouring, 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 Barát, Joret and Wood (2011), who proved that $c \ge \frac{4}{3}$ . Our lower-bound examples are obtained via the probabilistic method.

For a given list assignment $L\,=\,\{L\left( v \right)\,:\,v\,\in \,V\left( G \right)\}$, a graph $G\,=\,\left( V,\,E \right)$ is $L$-colorable if there exists a proper coloring $c$ of $G$ such that $c\left( v \right)\,\in \,L\left( v \right)$ for all $v\,\in \,V$. If $G$ is $L$-colorable for every list assignment $L$ having $\left| L\left( v \right) \right|\,\ge \,k$ for all $v\,\in \,V$, then $G$ is said to be $k$-choosable. Montassier (Inform. Process. Lett. 99 (2006) 68-71) conjectured that every planar graph without cycles of length 4, 5, 6, is 3-choosable. In this paper, we prove that every planar graph without 5-, 6- and 10-cycles, and without two triangles at distance less than 3 is 3-choosable.