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

TL;DR

This paper investigates the simple chromatic number of $(m,n)$-mixed graphs, showing polynomial-time computability for complete graphs and that reflexive targets do not reduce chromatic numbers for planar graphs and $k$-trees.

Contribution

It introduces the concept of simple chromatic number for $(m,n)$-mixed graphs and establishes complexity and structural results for specific graph families.

Findings

Polynomial-time computation for complete $(m,n)$-mixed graphs.

Reflexive targets do not lower chromatic number for planar graphs and $k$-trees.

Search for universal targets can be limited to simple cliques.

Abstract

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.