Neighbor sum distinguishing edge colorings of sparse graphs

Abstract

We consider proper edge colorings of a graph $G$ using colors of the set $\{1,\dots ,k\}$. Such a coloring is called neighbor sum distinguishing if for any $uv\in E\left(G\right)$, the sum of colors of the edges incident to $u$ is different from the sum of the colors of the edges incident to $v$. The smallest value of $k$ in such a coloring of $G$ is denoted by ${\mathrm{ndi}}_{\Sigma }\left(G\right)$. Let $\mathrm{mad}\left(G\right)$ and $\Delta \left(G\right)$ denote the maximum average degree and the maximum degree of a graph $G$, respectively. In this paper we show that, for a graph $G$ without isolated edges, if $\mathrm{mad}\left(G\right)<\frac{8}{3}$, then ${\mathrm{ndi}}_{\Sigma }\left(G\right)\le max\{\Delta \left(G\right)+1,7\}$; and if $\mathrm{mad}\left(G\right)<3$, then ${\mathrm{ndi}}_{\Sigma }\left(G\right)\le max\{\Delta \left(G\right)+2,7\}$.