Distant total irregularity strength of graphs via random vertex ordering

Abstract

Let c : V ∪ E → {1, 2, . . . , k} be a (not necessarily proper) total colouring of a graph G = (V, E) with maximum degree ∆. Two vertices u, v ∈ V are sum distinguished if they differ with respect to sums of their incident colours, i.e. c(u) + e∋u c(e) = c(v) + e∋v c(e). The least integer k admitting such colouring c under which every u, v ∈ V at distance 1 ≤ d(u, v) ≤ r in G are sum distinguished is denoted by ts r (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 + o(1))∆ r−1 for every integer r ≥ 2 and each graph G, thus improving the previously best result: ts r (G) ≤ 3∆ r−1 .Keywords: total vertex irregularity strength of a graph, 1-2 Conjecture, r-distant total irregularity strength of a graph