Distinguishing numbers and distinguishing indices of oriented graphs

TL;DR

This paper investigates the distinguishing numbers and indices of oriented graphs derived from basic graph classes, determining their extremal values across all orientations, and extends these concepts to directed graphs.

Contribution

It introduces and analyzes the parameters for oriented graphs, providing exact extremal values for various fundamental graph classes and extending the concepts to directed graphs.

Findings

Determined maximum and minimum distinguishing parameters for paths, cycles, complete graphs, and bipartite graphs.

Extended the notions of distinguishing numbers and indices to oriented graphs and studied their properties.

Provided bounds for unbalanced complete bipartite graphs with specific parameters.

Abstract

A distinguishing r-vertex-labelling (resp. r-edge-labelling) of an undirected graph G is a mapping $\lambda$ 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 D $\chi$ (G) and the distinguishing chromatic index D $\chi$ (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 value, taken over all possible orientations of the corresponding underlying graph, except for the minimum values for unbalanced complete bipartite graphs K m,n with m = 2, 3 or 4 and n > 3, 6 or 13, respectively, or m $\ge$ 5 and n > 2 m -- m 2 , for which we only provide upper bounds.