Asymptotically optimal neighbor sum distinguishing total colorings of graphs

Given a proper total k-coloring c : V(G) boolean OR E(G) -> {1, 2,, k} of a graph G, we define the value of a vertex v to be c(v) + Sigma C-uv is an element of E(G)(uv) The smallest integer k such that G has a proper total k-coloring whose values form a proper coloring is the neighbor sum distinguishing total chromatic number of G, chi(Sigma)'' Pilgniak and Woiniak (2013) conjectured that chi(Sigma)''(G) <= Delta(G) + 3 for any simple graph with maximum degree Delta(G). In this paper, we prove this bound to be asymptotically correct by showing that chi(Sigma)''(G) <= Delta(G)(1+ o(1)). The main idea of our argument relies on Przybylo's proof (2014) regarding neighbor sum distinguishing edge-colorings. (C) 2016 Elsevier B.V. All rights reserved.