A new upper bound for the total vertex irregularity strength of graphs

We investigate the following modification of the well known irregularity strength of graphs. Given a total weighting w of a graph G = (V,E) with elements of a set {1, 2, . . . , s}, denote wtG(v) = ∑ e3v w(e)+w(v) for each v ∈ V . The smallest s for which exists such a weighting with wtG(u) 6= wtG(v) whenever u and v are distinct vertices of G is called the total vertex irregularity strength of this graph, and is denoted by tvs(G). We prove that tvs(G) ≤ 3dn/δe+1 for each graph of order n and with minimum degree δ > 0.