Colourings of $(m, n)$-coloured mixed graphs

TL;DR

This paper generalizes existing results on vertex colourings of oriented and 2-edge-coloured graphs to a broader class called $(m, n)$-coloured mixed graphs, unifying various cases under a common framework.

Contribution

It introduces a unified approach to vertex colourings of $(m, n)$-coloured mixed graphs, extending and generalizing prior results such as Brooks' Theorem.

Findings

Results encompass and extend previous theorems for specific graph types.

Unified framework for vertex colourings of mixed graphs.

Connections to classical graph colouring theorems.

Abstract

A mixed graph is, informally, an object obtained from a simple undirected graph by choosing an orientation for a subset of its edges. A mixed graph is $(m, n)$-coloured if each edge is assigned one of $m \geq 0$ colours, and each arc is assigned one of $n \geq 0$ colours. Oriented graphs are $(0, 1)$-coloured mixed graphs, and 2-edge-coloured graphs are $(2, 0)$-coloured mixed graphs. We show that results of Sopena for vertex colourings of oriented graphs, and of Kostochka, Sopena and Zhu for vertex colourings oriented graphs and 2-edge-coloured graphs, are special cases of results about vertex colourings of $(m, n)$-coloured mixed graphs. Both of these can be regarded as a version of Brooks' Theorem.