Truncated degree DP-colourability of $K_{2,4}$-minor free graphs

Abstract

Assume $G$ is a graph and $k$ is a positive integer. Let $f$ from $V(G)$ to $ N$ be defined as $f(v)$ is the minimum of $k$ and $d(v)$. If $G$ is $f$-DP-colourable (respectively, $f$-choosable), then we say $G$ is $k$-truncated degree DP-colourable (respectively, $k$-truncated degree-choosable). Hutchinson proved that 2-connected maximal outerplanar graphs other than the triangle are $5$-truncated degree-choosable, and asked whether the result can be extended to all outerplanar graphs, and the question remained open. This paper proves that 2-connected $K24$-minor free graphs other than cycles and complete graphs are $5$-truncated degree DP-colourable. This not only answers Hutchinson's question in the affirmative, but also extends to a larger family of graphs, and strengthens choosability to DP-colourability.