Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs

A colouring of a graph $G=(V,E)$ is a function $c: V\rightarrow{1,2,\ldots }$ such that $c(u)\neq c(v)$ for every $uv\in E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $u\in V$, $L(u)$ is a subset of ${1, 2, \dots}$ of size $k$. A colouring $c$ of $G$ respects a $k$-regular list assignment $L$ of $G$ if $c(u)\in L(u)$ for every $u\in V$. A graph $G$ is $k$-choosable if for every $k$-regular list assignment $L$ of $G$, there exists a colouring of $G$ that respects $L$. We may also ask if for a given $k$-regular list assignment $L$ of a given graph $G$, there exists a colouring of $G$ that respects $L$. This yields the $k$-Regular List Colouring problem. For $k\in {3,4}$ we determine a family of classes ${\cal G}$ of planar graphs, such that either $k$-Regular List Colouring is NP-complete for instances $(G,L)$ with $G\in {\cal G}$, or every $G\in {\cal G}$ is $k$-choosable. By using known examples of non-$3$-choosable and non-$4$-choosable graphs, this enables us to classify the complexity of $k$-Regular List Colouring restricted to planar graphs, planar bipartite graphs, planar triangle-free graphs and to planar graphs with no $4$-cycles and no $5$-cycles. We also classify the complexity of $k$-Regular List Colouring and a number of related colouring problems for graphs with bounded maximum degree.