
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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Graph Labeling and Dimension Problems · Complexity and Algorithms in Graphs
