Variable degeneracy of planar graphs without chorded 6-cycles

Abstract

A cover of a graph $G$ is a graph $H$ with vertex set $V(H)={\bigcup }_{v\in V(G)}{L}_v$, where $L_v=\{v\}\times [s]$, and the edge set $M={\bigcup }_{uv\in E(G)}{M}_{uv}$, where $M_{uv}$ is a matching between $L_u$ and $L_v$. A vertex set $T\subseteq V(H)$ is a transversal of $H$ if $|T\cap L_v|=1$ for each $v\in V(G)$. Let $f$ be a nonnegative integer valued function on the vertex-set of $H$. If for any nonempty subgraph $\Gamma$ of $H[T]$, there exists a vertex $x\in V(H)$ such that $d(x)<f(x)$, then $T$ is called a strictly $f$-degenerate transversal. In this paper, we give a sufficient condition for the existence of strictly $f$-degenerate transversal for planar graphs without chorded $6$-cycles. As a consequence, every planar graph without subgraphs isomorphic to the configurations in Fig. 4 is DP-$4$-colorable.

16 pages, 13 figures