The complexity of planar graph choosability

Abstract

A graph G is k-choosable if for every assignment of a set S( v) of k colors to every vertex v of G, there is a proper coloring of G that assigns to each vertex v a color from S( v). We consider the complexity of deciding whether a given graph is k-choosable for some constant k. In particular, it is shown that deciding whether a given planar graph is 4-choosable is NP-hard, and so is the problem of deciding whether a given planar triangle-free graph is 3-choosable. We also obtain simple constructions of a planar graph which is not 4-choosable and a planar triangle-free graph which is not 3-choosable.