A note on the neighbour-distinguishing index of digraphs

TL;DR

This paper introduces a new concept of neighbour-distinguishing arc-colourings in directed graphs, establishing bounds on the minimum number of colours needed for various classes of digraphs.

Contribution

It defines the neighbour-distinguishing index for digraphs and provides upper bounds for this index across different graph classes, advancing graph colouring theory.

Findings

Upper bounds on neighbour-distinguishing index for various digraph classes

Introduction of a new arc-colouring concept for digraphs

Formalization of neighbour-distinguishing arc-colouring criteria

Abstract

In this note, we introduce and study a new version of neighbour-distinguishing arc-colourings of digraphs. An arc-colouring $\gamma$ 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 $S_\gamma^-(u)$ and $S_\gamma^+(u)$ the sets of colours that appear on the incoming arcs and on the outgoing arcs of $u$, respectively. An arc colouring $\gamma$ of $D$ is \emph{neighbour-distinguishing} if, for every two adjacent vertices $u$ and $v$ of $D$, the ordered pairs $(S_\gamma^-(u),S_\gamma^+(u))$ and $(S_\gamma^-(v),S_\gamma^+(v))$ 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.