A local condition for planar graphs to be 4-choosable

Abstract

For a given list assignment $L$, a graph $G$ is list $L$-colorable if there exists a proper coloring $c$ of the vertices such that $c(v)\in~L(v)$ for all $v\in~V(G)$. If $G$ is list $L$-colorable for every list assignment $L$ with $|L(v)|\geq~k$ for all $v\in~V(G)$, then $G$ is $k$-choosable. In this paper, we give a local condition for planar graphs to be 4-choosable, i.e., we show that if $G$ is a planar graph having no vertex incident to 3-, 4-, 5-, 6-cycles simultaneously, then $G$ is 4-choosable.