Geodesic
Orientations and $3$-colourings of graphs
Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 3, pp. 549-553.

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We provide the list of all paths with at most $16$ arcs with the property that if a graph $G$ admits an orientation $\vec{G}$ such that one of the paths in our list admits no homomorphism to $\vec{G}$, then $G$ is $3$-colourable.
Classification : 05C15, 05C20
Keywords: graph colourings; paths; Hasse-Gallai-Roy Theorem 
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     title = {Orientations and $3$-colourings of graphs},
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Chouinard-Prévost, Vincent; Côté, Alexandre; Tardif, Claude. Orientations and $3$-colourings of graphs. Commentationes Mathematicae Universitatis Carolinae, Tome 45 (2004) no. 3, pp. 549-553. http://geodesic.mathdoc.fr/item/CMUC_2004__45_3_a15/