Distinguishing arc-colourings of symmetric digraphs

Abstract

A symmetric digraph is obtained from an undirected graph by replacing each edge uv of by a pair of opposite arcs and . An arc-colouring of a digraph is called distinguishing if the only automorphism preserving it is the identity. The least number of colours in a distinguishing arc-colouring, not necessarily proper, of is called the distinguishing index D'(). We study bounds for D'(). For proper distinguishing arc-colourings, the least number of colours is called the distinguishing chromatic index of . There are 15 possible types of proper arc-colourings of a digraph depending on the definition of adjacent arcs. In this paper we investigate distinguishing chromatic indices of for the nine remaining types not considered in our two previous papers. We formulate several conjectures.