Edge-Choosability of Planar Graphs Without Chordal $7$-Cycles

Abstract

A graph $G$ is edge-$L$-colorable, if for a given edge assignment $L=\{L(e):e\in E(G)\}$, there exists a proper edge-coloring $\varphi$ of $G$ such that $\varphi (e)\in L(e)$ for all $e\in E(G)$. If $G$ is edge-$L$-colorable for every edge assignment $L$ with $|L(e)|\ge k$ for $e\in E(G)$, then $G$ is said to be edge-$k$-choosable. In this paper, we prove that if $G$ is a planar graph without chordal $7$-cycles, then $G$ is edge-$k$-choosable, where $k=max\{8,\mathrm{\Delta }(G)+1\}$.