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

A proper total $k$-colouring of a graph $G=(V,E)$ is an assignment $c : V \cup E\to \{1,2,\ldots,k\}$ of colours to the edges and the vertices of $G$ such that no two adjacent edges or vertices and no edge and its end-vertices are associated with the same colour. A total neighbour sum distinguishing $k$-colouring, or tnsd $k$-colouring for short, is a proper total $k$-colouring such that $\sum_{e\ni u}c(e)+c(u)\neq \sum_{e\ni v}c(e)+c(v)$ for every edge $uv$ of $G$. We denote by $\chi''_{\Sigma}(G)$ the total neighbour sum distinguishing index of $G$, which is the least integer $k$ such that a tnsd edge $k$-colouring of $G$ exists. It has been conjectured that $\chi''_{\Sigma}(G) \leq \Delta(G) + 3$ for every graph $G$. In this paper we confirm this conjecture for any graph $G$ with ${\rm mad}(G)<\frac{14}{3}$ and $\Delta(G) \geq 8$.