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

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 $m\ge 0$ colours, and each arc is assigned one of $n\ge 0$ 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.

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