Geodesic
An analogue of quasi-transitivity for edge-coloured graphs
Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 3, pp. 1189-1215

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We extend the notion of quasi-transitive orientations of graphs to 2-edge-coloured graphs. By relating quasi-transitive 2-edge-colourings to an equivalence relation on the edge set of a graph, we classify those graphs that admit a quasi-transitive 2-edge-colouring. As a contrast to Ghouilá-Houri's classification of quasi-transitively orientable graphs as comparability graphs, we find quasi-transitively 2-edge-colourable graphs do not admit a forbiddden subgraph characterization. Restricting the problem to comparability graphs, we show that the family of uniquely quasi-transitively orientable comparability graphs is exactly the family of comparabilty graphs that admit no quasi-transitive 2-edge-colouring.
Keywords: oriented graph, quasi-transitivity, edge colouring, uniquely quasi-transitively colourable graphs 
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Duffy, Christopher; Mullen, Todd. An analogue of quasi-transitivity for edge-coloured graphs. Discussiones Mathematicae. Graph Theory, Tome 44 (2024) no. 3, pp. 1189-1215. http://geodesic.mathdoc.fr/item/DMGT_2024_44_3_a18/