A note on asymptotically optimal neighbour sum distinguishing colourings

The least $k$ admitting a proper edge colouring $c:E\to \{1,2,\dots ,k\}$ of a graph $G=(V,E)$ without isolated edges such that $\sum_{e\ni u}c(e)\neq \sum_{e\ni v}c(e)$forevery$uv\in E$isdenotedby$\chi’{\Sigma}(G)$. It has been conjectured that ${\chi }_{\mathrm{\Sigma }}^{\mathrm{\prime }}(G)\le \mathrm{\Delta }+2$ for every connected graph of order at least three different from the cycle $C_5$, where $\mathrm{\Delta }$ is the maximum degree of $G$. It is known that $\chi’{\Sigma}(G) = \Delta + O(\Delta^\frac{5}{6}\ln^\frac{1}{6}\Delta)$foragraph$G$ 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.