Weak$^*$ degeneracy and weak$^*$ $k$-truncated-degree-degenerate graphs

Huan Zhou, Jialu Zhu, Xuding Zhu

Published 2023-08-30Version 1

This paper introduces the concept of weak$^*$ degeneracy of a graph that shares many nice properties of degeneracy. We prove that for any $f: V(G) \to \mathbb{N}$, if $G$ is weak$^*$ $f$-degenerate, then $G$ is DP-$f$-paintable and $f$-AT. Hence the weak$^*$ degeneracy of $G$ is an upper bound for many colouring parameters, including the online DP-chromatic number and the Alon-Tarsi number. Let $k$ be a positive integer and let $f(v)=\min\{d_G(v), k\}$ for each vertex $v$ of $G$. If $G$ is weak$^*$ $f$-degenerate (respectively, $f$-choosable), then we say $G$ is weak$^*$ $k$-truncated-degree--degenerate (respectively, $k$-truncated-degree-choosable). Richtor asked whether every 3-connected non-complete planar graph is $6$-truncated-degree-choosable. We construct a 3-connected non-complete planar graph which is not $7$-truncated-degree-choosable, so the answer to Richtor's question is negative even if 6 is replaced by 7. Then we prove that every 3-connected non-complete planar graph is weak$^*$ $16$-truncated-degree-degenerate (and hence $16$-truncated-degree-choosable). 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 $t$. We prove that there is a constant $k$ such that every $s$-connected non-complete graph $G \in {\mathcal G}$ is weak$^*$ $k$-truncated-degree-degenerate. In particular, for any surface $\Sigma$, there is a constant $k$ such that every 3-connected non-complete graph embeddable on $\Sigma$ is weak$^*$ $k$-truncated-degree-degenerate.