A note on the neighbour-distinguishing index of digraphs

09/23/2019

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by   Eric Sopena, et al.

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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.