The Simple Chromatic Number of $(m,n)$-Mixed Graphs

An $(m,n)$-mixed graph generalizes the notions of oriented graphs and edge-coloured graphs to a graph object with $m$ arc types and $n$ edge types. A simple colouring of such a graph is a non-trivial homomorphism to a reflexive target. We find that simple chromatic number of complete $(m,n)$-mixed graphs can be found in polynomial time. For planar graphs and $k$-trees ($k \geq 3$) we find that allowing the target to be reflexive does not lower the chromatic number of the respective family of $(m,n)$-mixed graphs. This implies that the search for universal targets for such families may be restricted to simple cliques.