Colourings of (m, n)-coloured mixed graphs

10/18/2020

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by   Gary MacGillivray, et al.

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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 ≥ 0 colours, and each arc is assigned one of n ≥ 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.