An Analogue of Quasi-Transitivity for Edge-Coloured Graphs

TL;DR

This paper extends the concept of quasi-transitive orientations to 2-edge-coloured graphs, classifies those graphs that admit such colourings, and explores their properties and restrictions within comparability graphs.

Contribution

It introduces a new extension of quasi-transitivity to 2-edge-coloured graphs and provides a classification and characterization of these graphs, contrasting with existing results.

Findings

Quasi-transitive 2-edge-colourings relate to an equivalence relation on edges.

Such graphs do not have a forbidden subgraph characterization.

Uniquely quasi-transitively orientable comparability graphs are those with no quasi-transitive 2-edge-colouring.

Abstract

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 Ghouil\'{a}-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.