Degree-truncated choosability of graphs

Abstract

A graph $G$ is called degree-truncated $k$-choosable if for every list assignment $L$ with $|L(v)| \ge \min\{d_G(v), k\}$ for each vertex $v$, $G$ is $L$-colourable. Richter asked whether every 3-connected non-complete planar graph is degree-truncated 6-choosable. We answer this question in negative by constructing a 3-connected non-complete planar graph which is not degree-truncated 7-choosable. Then we prove that every 3-connected non-complete planar graph is degree-truncated 16-DP-colourable (and hence degree-truncated $16$-choosable). We further prove that for an arbitrary proper minor closed family ${\mathcal G}$ of graphs, let $s$ be the minimum integer such that $K_{s,t} \notin \mathcal{G}$ for some $t$, then there is a constant $k$ such that every $s$-connected graph $G \in {\mathcal G}$ other than a GDP tree is degree-truncated DP-$k$-colourable (and hence degree-truncated $k$-choosable), where a GDP-tree is a graph whose blocks are complete graphs or cycles. In particular, for any surface $\Sigma$, there is a constant $k$ such that every 3-connected non-complete graph embeddable on $\Sigma$ is degree-truncated DP-$k$-colourable (and hence degree-truncated $k$-choosable). The $s$-connectedness for graphs in $\mathcal{G}$ (and 3-connectedness for graphs embeddable on $\Sigma$) is necessary, as for any positive integer $k$, $K_{s-1,k^{s-1}} \in \mathcal{G}$ ($K_{2,k^2}$ is planar) is not degree-truncated $k$-choosable. Also, non-completeness is a necessary condition, as complete graphs are not degree-choosable.