Distant total irregularity strength of graphs via random vertex ordering

Let c : V boolean OR E -> {1, 2, ..., k} be a (not necessarily proper) total colouring of a graph G = (V, E) with maximum degree d. Two vertices u, v is an element of V are sum distinguished if they differ with respect to sums of their incident colours, i.e. c(u)+ Sigma(e(sic)u) c(e) not equal c(v)+Sigma(e(sic)u) c(e). The least integer k admitting such colouring c under which every u, v E V at distance 1 <= d(u, v) <= r in G are sum distinguished is denoted by tsr(G). Such graph invariants link the concept of the total vertex irregularity strength of graphs with so-called 1-2-Conjecture, whose concern is the case of r = 1. Within this paper we combine probabilistic approach with purely combinatorial one in order to prove that ts(r)(G) <= (2 + 0(1))Delta(r-1) for every integer r >= 2 and each graph G, thus improving the previously best result: ts(r)(G) <= 3 Delta(r-1) (C) 2017 Elsevier B.V. All rights reserved.