Geodesic
On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets
Discrete mathematics & theoretical computer science, Tome 23 (2021-2022) no. 1.

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We consider non-trivial homomorphisms to reflexive oriented graphs in which some pair of adjacent vertices have the same image. Using a notion of convexity for oriented graphs, we study those oriented graphs that do not admit such homomorphisms. We fully classify those oriented graphs with tree-width $2$ that do not admit such homomorphisms and show that it is NP-complete to decide if a graph admits an orientation that does not admit such homomorphisms. We prove analogous results for $2$-edge-coloured graphs. We apply our results on oriented graphs to provide a new tool in the study of chromatic number of orientations of planar graphs -- a long-standing open problem.
DOI : 10.46298/dmtcs.6773
Classification : 05C15, 05C60 
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Duffy, Christopher; Shan, Sonja Linghui. On the existence and non-existence of improper homomorphisms of oriented and $2$-edge-coloured graphs to reflexive targets. Discrete mathematics & theoretical computer science, Tome 23 (2021-2022) no. 1. doi : 10.46298/dmtcs.6773. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6773/

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