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 $c : V \cup E\to {1,2,\ldots,k}$ofcolourstotheedgesandtheverticesof$G$ 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 ${\sum }_{e\ni u}c(e)+c(u)\mathrm{\ne }{\sum }_{e\ni v}c(e)+c(v)$ for every edge $uv$ of $G$. We denote by ${\chi }_{\mathrm{\Sigma }}^{\mathrm{\prime }\mathrm{\prime }}(G)$ 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 ${\chi }_{\mathrm{\Sigma }}^{\mathrm{\prime }\mathrm{\prime }}(G)\le \mathrm{\Delta }(G)+3$ for every graph $G$. In this paper we confirm this conjecture for any graph $G$ with $\mathrm{m}\mathrm{a}\mathrm{d}(G)<\frac{14}{3}$ and $\mathrm{\Delta }(G)\ge 8$.