Distinguishing numbers and distinguishing indices of oriented graphs

Kahina Meslem (1) , Eric Sopena (2)

1 Laboratoire de Recherche Opérationnelle et Mathématiques de la Décision - LAROMAD (Alger, Algérie)
2 LaBRI - Laboratoire Bordelais de Recherche en Informatique

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 D χ (G) and the distinguishing chromatic index D χ (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 ≥ 5 and n > 2 m − m 2 , for which we only provide upper bounds.

Mots clés: Distinguishing chromatic index; Distinguishing chromatic number; Distinguishing number; Distinguishing index; Automorphism group; Oriented graph; Complete bipartite graph

Domaines: Combinatoire [math.CO]; Mathématique discrète [cs.DM]

Kahina Meslem, Eric Sopena. Distinguishing numbers and distinguishing indices of oriented graphs. Discrete Applied Mathematics, 2020, 285, pp.330-342. ⟨10.1016/j.dam.2020.06.007⟩. ⟨hal-02334616v2⟩