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

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 ${\rm mad}(G)<3$, $\Delta(G) \geq 6$.