Brooks-type colourings of digraphs in linear time

TL;DR

The paper extends Brooks-type colourings to digraphs by introducing variable bidegeneracy and the notion of $F$-dicolouring, where each colour class induces a strictly-$f_i$-bidegenerate subdigraph. It defines hard pairs as the only obstructions to $F$-dicolourability and proves a Brooks-type theorem: a connected digraph $D$ is $F$-dicolourable if and only if $(D,F)$ is not a hard pair. The authors develop a comprehensive linear-time algorithm that either constructs an $F$-dicolouring or certifies nonexistence, using block contractions, safe colourings, and a two-colour reduction with CSP-based ear-decompositions. This yields the first linear-time algorithms for several generalisations of Brooks’ theorem for (di)graphs, and provides a unified framework that also yields linear-time solutions for list-dicolouring and related degenerate-partition variants, with practical implications for efficient colourings in directed graphs.

Abstract

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 $F = (f_1,...,f_s)$ 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.

Figures (3)

• Figure 1: A digraph which is strictly-$(2,0)$-bidegenerate while it is not strictly-$(0,2)$-bidegenerate.
• Figure 2: An example of a hard pair $(D,F)$ where $F$ is a sequence of three functions $f_1,f_2,f_3$ the values of which are respectively represented in red, green, and blue. For the sake of readability, the value of $f_i$ is missing when it is equal to $(0,0)$.
• Figure 5: Ears $H_j$ and $H_k$ recombined into $H'$ (light blue) and $H"$ (pink), each of length at least two. On the left, the case where $H_k$ has one endpoint in $H_j$. In the middle, the case where $H_k$ has its two endpoints in $H_j$. On the right, the case where $H_k$ has its two endpoints in the cycle $H_j=H_0$.

Theorems & Definitions (51)

• Theorem 1: Brooks brooksMPCPS37
• Theorem 2: Borodin borodinThesis ; Erdős, Rubin, Taylor erdosCN26
• Theorem 3: Bollobás and Manvel bollobasBLMS11 ; Borodin borodinDA28
• Theorem 4: Borodin, Kostochka, Toft borodinDM214
• Theorem 5: Mohar moharLAA432
• Theorem 6: Harutyunyan and Mohar harutyunyanSIDMA25
• Theorem 7: Bang-Jensen, Schweser, Stiebitz bangSIDMA36
• Theorem 8
• Lemma 9
• proof