On the neighbour sum distinguishing index of graphs with bounded maximum average degree

TL;DR

This paper proves that for graphs with maximum average degree less than 3 and minimum degree at least 6, the neighbour sum distinguishing index is at most one more than the maximum degree, improving understanding of edge colourings.

Contribution

It establishes an upper bound of (G)+1 for the neighbour sum distinguishing index in graphs with bounded maximum average degree and minimum degree at least 6.

Findings

(G)+1 bound proven for specific graph class

Graphs with mad(G)<3 and (G)6 have neighbour sum distinguishing index at most (G)+1

Advances the understanding of edge colourings in sparse graphs

Abstract

A proper edge $k$-colouring of a graph $G=(V,E)$ is an assignment $c:E\to \{1,2,\ldots,k\}$ of colours to the edges of the graph such that no two adjacent edges are associated with the same colour. A neighbour sum distinguishing edge $k$-colouring, or nsd $k$-colouring for short, is a proper edge $k$-colouring such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every edge $uv$ of $G$. We denote by $\chi'_{\sum}(G)$ the neighbour sum distinguishing index of $G$, which is the least integer $k$ such that an nsd $k$-colouring of $G$ exists. By definition at least maximum degree, $\Delta(G)$ colours are needed for this goal. In this paper we prove that $\chi'_\Sigma(G) \leq \Delta(G)+1$ for any graph $G$ without isolated edges and with ${\rm mad}(G)<3$, $\Delta(G) \geq 6$.