An analogue of quasi-transitivity for edge-coloured graphs

Abstract

We extend the notion of quasi-transitive orientations of graphs to 2edge-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.

Chromatic Polynomials of 2-Edge-Coloured Graphs

“…The classification of chromatically invariant oriented graphs given in [7] bears little resemblance to the statement of Theorem 16. This is due to the fact that the family of graphs that admit an orientation with no induced 2-dipath differs drastically from the family that admit a non-trivial 2-edge-colouring with no induced bichromatic 2-path (see [9] and [8])…”