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This paper introduces the concept of weak$^*$ degeneracy of a graph thatshares many nice properties of degeneracy. Using this concept, we derive upperbounds for many colouring parameter. In particular, combined with an earlierresult, the upper bounds imply that planar graphs of girth 5 are 3-AT, whichstrengthens Thomassen's result that planar graphs of girth 5 are 3-choosable.Let $k$ be a positive integer and let $f(v)=\min\{d_G(v), k\}$ for each vertex$v$ of $G$. If $G$ is $f$-choosable, then we say $G$ is$k$-truncated-degree-choosable. Richtor asked whether every 3-connectednon-complete planar graph is $6$-truncated-degree-choosable. We construct a3-connected non-complete planar graph which is not$7$-truncated-degree-choosable, so the answer to Richtor's question is negativeeven if 6 is replaced by 7. Then we prove that every 3-connected non-completeplanar graph is $16$-truncated-degree-choosable. For an arbitrary proper minorclosed family ${\mathcal G}$ of graphs, let $s$ be the minimum integer suchthat $K_{s,t} \notin \mathcal{G}$ for some $t$. We prove that there is aconstant $k$ such that every $s$-connected non-complete graph $G \in {\mathcalG}$ is $k$-truncated-degree-choosable.  In particular, for any surface $\Sigma$, there is a constant $k$ such thatevery 3-connected non-complete graph embeddable on $\Sigma$ is$k$-truncated-degree-choosable. The choosability results are proved viaweak$^*$-degeneracy, and hence remain true when choosable is replaced byDP-colourable or paintable or DP-paintable or AT.

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

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