Colourings of $(m, n)$-coloured mixed graphs
Gary MacGillivray, Shahla Nasserasr, Feiran Yang
TL;DR
The paper develops a unified framework for vertex colourings of $(m,n)$-coloured mixed graphs, extending and connecting results for oriented graphs and 2-edge-coloured graphs through the chromatic number $\chi(G, m, n)$. It presents two Brooks-type results: a constructive bound using a family of universal graphs $\mathcal{Z}_{m,n,2\Delta-1}$ that yields $\chi(G, m, n) \le (2\Delta-1)(m+2n)^{2\Delta-2}$, and a probabilistic bound showing $\chi(G, m, n) \le \Delta^2(m+2n)^{\Delta+1}$ for $k \ge 4$, both generalizing known bounds for special cases. The work introduces and leverages the concept of $k$-hom-universal graphs and the extension-control Property $P_{a,b}$, linking these colourings to homomorphisms and providing a framework that encompasses and extends previous theorems by Sopena and by Kostochka, Sopena, and Zhu. Together, the constructive and probabilistic approaches offer complementary tools for bounding the $(m,n)$-coloured mixed chromatic number across diverse graph families.
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.