Arc-distinguishing of orientations of graphs

Abstract

The distinguishing index $D^{\prime }(G)$ of a graph $G$ is the minimum number of colours in an edge colouring preserved only by the identity automorphism. We study how orienting the edges affects this parameter, relating the minimum and maximum distinguishing indices over all orientations of $G$ to $D^{\prime }(G)$. We establish bounds and exact relations for bipartite graphs and trees, and study rigid orientations of traceable and claw-free graphs. Our results answer a question of Meslem and Sopena on unbalanced complete bipartite graphs.