Brooks-Type Colourings of Digraphs in Linear Time

Brooks' Theorem is a fundamental result on graph colouring, stating that the chromatic number of a graph is almost always upper bounded by its maximal degree. Lovász showed that such a colouring may then be computed in linear time when it exists. Many analogues are known for variants of (di)graph colouring, notably for list-colouring and partitions into subgraphs with prescribed degeneracy. One of the most general results of this kind is due to Borodin, Kostochka, and Toft, when asking for classes of colours to satisfy ‘variable degeneracy’ constraints. An extension of this result to digraphs has recently been proposed by Bang-Jensen, Schweser, and Stiebitz, by considering colourings as partitions into ‘variable weakly degenerate’ subdigraphs. Unlike earlier variants, there exists no linear-time algorithm to produce colourings for these generalisations. We introduce the notion of (variable) bidegeneracy for digraphs, capturing multiple (di)graph degeneracy variants. We define the corresponding concept of $F$-dicolouring, where $F=\left(f_1,\dots ,f_s\right)$ is a vector of functions, and an $F$-dicolouring requires vertices coloured $i$ to induce a ‘strictly-$f_i$-bidegenerate’ subdigraph. We prove an analogue of Brooks' theorem for $F$-dicolouring, generalising the result of Bang-Jensen et al., and earlier analogues in turn. Our new approach provides a linear-time algorithm that, given a digraph $D$, either produces an $F$-dicolouring of $D$, or correctly certifies that none exist. This yields the first linear-time algorithms to compute (di)colourings corresponding to the aforementioned generalisations of Brooks' theorem. In turn, it gives an unified framework to compute such colourings for various intermediate generalisations of Brooks' theorem such as list-(di)colouring and partitioning into (variable) degenerate sub(di)graphs.