Distinguishing numbers and distinguishing indices of oriented graphs

Abstract

A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected graph G is a mapping λ from the set of vertices (resp. the set of edges) of G to the set of labels {1,…,r} such that no non-trivial automorphism of G preserves all the vertex (resp. edge) labels. The distinguishing number D(G) and the distinguishing index D′(G) of G are then the smallest r for which G admits a distinguishing r-vertex-labelling or r-edge-labelling, respectively. The distinguishing chromatic number Dχ(G) and the distinguishing chromatic index Dχ′(G) are defined similarly, with the additional requirement that the corresponding labelling must be a proper colouring.These notions readily extend to oriented graphs, by considering arcs instead of edges. In this paper, we study the four corresponding parameters for oriented graphs whose underlying graph is a path, a cycle, a complete graph or a bipartite complete graph. In each case, we determine their minimum and maximum values, taken over all possible orientations of the corresponding underlying graph, except for the minimum values for unbalanced complete bipartite graphs Km,n with m=2, 3 or 4 and n>3, 6 or 13, respectively, or m≥5 and n>2m−m2, for which we only provide upper bounds.