Distant total sum distinguishing index of graphs

Abstract

Let c:V∪E→{1,2,…,k} be a proper total colouring of a graph G=(V,E) with maximum degree Δ. We say vertices u,v∈V are sum distinguished if c(u)+∑e∋uc(e)≠c(v)+∑e∋vc(e). By χΣ,r′′(G) we denote the least integer k admitting such a colouring c for which every u,v∈V, u≠v, at distance at most r from each other are sum distinguished in G. For every positive integer r an infinite family of examples is known with χΣ,r′′(G)=Ω(Δr−1). In this paper we prove that χΣ,r′′(G)≤(2+o(1))Δr−1 for every integer r≥3 and each graph G, while χΣ,2′′(G)≤(18+o(1))Δ.