A note on asymptotically optimal neighbour sum distinguishing colourings

Abstract

The least $k$ admitting a proper edge colouring $c:E\to\{1,2,\ldots,k\}$ of a graph $G=(V,E)$ without isolated edges such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$ for every $uv\in E$ is denoted by $\chi'_{\Sigma}(G)$. It has been conjectured that $\chi'_{\Sigma}(G)\leq \Delta + 2$ for every connected graph of order at least three different from the cycle $C_5$, where $\Delta$ is the maximum degree of $G$. It is known that $\chi'_{\Sigma}(G) = \Delta + O(\Delta^\frac{5}{6}\ln^\frac{1}{6}\Delta)$ for a graph $G$ without isolated edges. We improve this upper bound to $\chi'_{\Sigma}(G) = \Delta + O(\Delta^\frac{1}{2})$ using a simpler approach involving a combinatorial algorithm enhanced by the probabilistic method. The same upper bound is provided for the total version of this problem as well.