Colouring Digraphs

TL;DR

This thesis explores how digraph structure influences the dichromatic number, extending classical graph coloring results to directed graphs, and addresses related conjectures and edge-coloring problems with new bounds and partial solutions.

Contribution

It extends Brooks' theorem and related degree bounds to digraphs, proves subcases of the Gyárfás-Sumner conjecture for digraphs, and generalizes edge-coloring bounds for multigraphs with defective coloring.

Findings

Directed Brooks-like theorems depend on notions of maximum directed degree.

Several subcases of the directed Gyárfás-Sumner conjecture are proved.

Bounds for d-edge-defective coloring are extended to all d, generalizing Vizing's theorem.

Abstract

The aim of this thesis is to investigate how the structure of a digraph affects its dichromatic number and to extend various results on undirected colouring to digraphs. In the first part of this thesis, we examine how the dichromatic number interacts with other metrics. First, we consider the degree, which is the maximum number of neighbours of a vertex. In the undirected case, this corresponds to Brooks' theorem, a celebrated theorem with multiple variations and generalizations. In the directed case, there is no natural metric corresponding to the maximum degree, so we explore how different notions of maximum directed degree lead to either Brooks-like theorems or impossibility results. We also investigate the maximum local-arc connectivity, a metric that encompasses several degree-like metrics. The second part of this manuscript focuses on a directed analogue of the Gy\'arf\'as-Sumner conjecture. The Gy\'arf\'as-Sumner conjecture tries to characterize sets S of undirected graphs such that graphs with large enough chromatic number must contain a graph of S. This conjecture is still largely open. On digraphs, a corresponding conjecture was proposed by Aboulker, Charbit, and Naserasr. We prove several subcases of this conjecture, mainly demonstrating that certain classes of digraphs have bounded dichromatic number. In the last part of this thesis, we address the d-edge-defective-colouring problem, which involves colouring edges of a multigraph such that, for any vertex, no colour appears on more than d of its incident edges. When d equals one, this corresponds to the infamous edge-colouring problem. Shannon established a tight bound on the number of colours needed relative to the maximum degree when d equals one, and we extend this result to any value of d. We also explore this problem on simple graphs and prove results that extend Vizing's theorem to any value of d.