Filling the complexity gaps for colouring planar and bounded degree graphs

Abstract

AbstractA colouring of a graph is a function such that for every . A ‐regular list assignment of is a function with domain such that for every , is a subset of of size . A colouring of respects a ‐regular list assignment of if for every . A graph is ‐choosable if for every ‐regular list assignment of , there exists a colouring of that respects . We may also ask if for a given ‐regular list assignment of a given graph , there exists a colouring of that respects . This yields the ‐Regular List Colouring problem. For , we determine a family of classes of planar graphs, such that either ‐Regular List Colouring is ‐complete for instances with , or every is ‐choosable. By using known examples of non‐‐choosable and non‐‐choosable graphs, this enables us to classify the complexity of ‐Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle‐free graphs, and planar graphs with no ‐cycles and no ‐cycles. We also classify the complexity of ‐Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.