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 ${\sum }_{e\ni u}c(e)\mathrm{\ne }{\sum }_{e\ni v}c(e)$ for every edge $uv$ of $G$. We denote by ${\chi }_{\sum }^{\mathrm{\prime }}(G)$ 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, $\mathrm{\Delta }(G)$ colours are needed for this goal. In this paper we prove that $\chi’_\Sigma(G) \leq \Delta(G)+1$foranygraph$G$withoutisolatededgesandwith${\rm mad}(G)<3$,$\Delta(G) \geq 6$.