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

Abstract

A proper total $k$-colouring of a graph $G=(V,E)$ is an assignment $c:V\cup E\to \{1,2,\dots ,k\}$ 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 ${\sum }_{e\ni u}c(e)+c(u)\ne {\sum }_{e\ni v}c(e)+c(v)$ for every edge $uv$ of $G$. We denote by ${\chi }_{\Sigma }^{\prime \prime }(G)$ 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 ${\chi }_{\Sigma }^{\prime \prime }(G)\le \Delta (G)+3$ for every graph $G$. In this paper we confirm this conjecture for any graph $G$ with $\mathrm{mad}(G)<\frac{14}{3}$ and $\Delta (G)\ge 8$.

arXiv comment10 pages. arXiv admin note: text overlap with arXiv:1508.06112