A note on asymptotically optimal neighbour sum distinguishing colourings

The least k admitting a proper edge colouring c : E -> {1, 2, ..., k} of a graph G = (V, E) without isolated edges such that Sigma(e(sic)u) c(e) not equal Sigma(e(sic)v) c (e) for every uv is an element of E is denoted by chi(Sigma')(G). It has been conjectured that chi(Sigma')(G) <= 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(5/6) ln(1/6) Delta) for a graph G without isolated edges. We improve this upper bound to chi(Sigma')(G) = Delta + O(Delta(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. (C) 2018 Elsevier Ltd. All rights reserved.