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 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 (Δ
r− 1). In this paper we prove that χ Σ, r′′(G)≤(2+ o (1)) Δ r− 1 for every integer r≥ 3 and …