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 ∆. We say vertices u, v ∈ V are sum distinguished if c(u)+ e∋u c(e) = c(v) + e∋v c(e). By χ ′′ Σ,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 χ ′′ Σ,r (G) = Ω(∆ r−1 ). In this paper we prove that χ ′′ Σ,r (G) ≤ (2 + o(1))∆ r−1 for every integer r ≥ 3 and each graph G, while χ ′′ Σ,2 (G) ≤ (18 + o(1))∆.