Every Plane Graph with Girth at least $4$ without $8$- and $9$-circuits is $3$-Choosable

Abstract

The choice number of a graph $G$, denoted by ${\chi }_l(G)$, is the minimum number ${\chi }_l$ such that if we give lists of ${\chi }_l$ colors to each vertex of $G$, there is a vertex coloring of $G$ where each vertex receives a color from its own list no matter what the lists are. In this paper, we show that ${\chi }_l(G)\le 3$ for each plane graph of girth at least $4$ which contains no $8$-circuits and $9$-circuits.