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 \(\chi_l(G) \leq 3\) for each plane graph of girth at least \(4\) which contains no \(8\)-circuits and \(9\)-circuits.