Facial list colourings of plane graphs

Abstract

Let G=(V,E,F) be a connected plane graph, with vertex set V, edge set E, and face set F. For X∈{V,E,F,V∪E,V∪F,E∪F,V∪E∪F}, two distinct elements of X are facially adjacent in G if they are incident elements, adjacent vertices, adjacent faces, or facially adjacent edges (edges that are consecutive on the boundary walk of a face of G). A list k-colouring is facial with respect to X if there is a list k-colouring of elements of X such that facially adjacent elements of X receive different colours. We prove that every plane graph G=(V,E,F) has a facial list 4-colouring with respect to X=E, a facial list 6-colouring with respect to X∈{V∪E,E∪F}, and a facial list 8-colouring with respect to X=V∪E∪F. For plane triangulations, each of these results is improved by one and it is tight. These results complete the theorem of Thomassen that every plane graph has a (facial) list 5-colouring with respect to X∈{V,F} and the theorem of Wang and Lih that every simple plane graph has a (facial) list 7-colouring with respect to X=V∪F.