A note on the neighbour-distinguishing index of digraphs

Abstract

In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring γ of a digraph D is proper if no two arcs with the same head or with the same tail are assigned the same colour. For each vertex u of D, we denote by S_γ⁻(u) and S_γ⁺(u) the sets of colours that appear on the incoming arcs and on the outgoing arcs of u, respectively. An arc colouring γ of D is neighbour-distinguishing if, for every two adjacent vertices u and v of D, the ordered pairs (S_γ⁻(u), S_γ⁺(u)) and (S_γ⁻(v), S_γ⁺(v)) are distinct. The neighbour-distinguishing index of D is then the smallest number of colours needed for a neighbour-distinguishing arc-colouring of D.

We prove upper bounds on the neighbour-distinguishing index of various classes of digraphs.