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• W751048960 abstract "A colouring of a graph $G=(V,E)$ is a function $c: Vrightarrow{1,2,ldots }$ such that $c(u)neq c(v)$ for every $uvin E$. A $k$-regular list assignment of $G$ is a function $L$ with domain $V$ such that for every $uin V$, $L(u)$ 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 $uin V$. 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 $kin {3,4}$ 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 $Gin {cal G}$, or every $Gin {cal G}$ 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." @default.
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• W751048960 date "2015-06-22" @default.
• W751048960 modified "2023-09-25" @default.
• W751048960 title "Filling the Complexity Gaps for Colouring Planar and Bounded Degree Graphs" @default.
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