Proper distinguishing arc-colourings of symmetric digraphs

Abstract

A symmetric digraph G↔ arises from a simple graph G by substituting each edge uv by a pair of opposite arcs uv→,vu→. An arc-colouring c of G↔ is distinguishing if the only automorphism of G↔ preserving c is the identity. We study four types of proper arc-colourings of G↔ corresponding to four definitions of adjacency of arcs. For each type, we investigate the distinguishing chromatic index of G↔, i.e. the least number of colours in a distinguishing proper colouring of G↔. We also determine tight bounds for chromatic indices of G↔, i.e. for the least numbers of colours in each type of proper colourings. Colourings of arcs of a symmetric digraph G↔ are equivalent to colourings of halfedges of the graph G, which have applications in computer science.