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 w t G ( v ) = ∑ e ∋ v w ( e ) + w ( v ) for each v ∈ V . The smallest s for which exists such a weighting with w t G ( u ) ≠ w t G ( 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 ) ≤ 3 ⌈ n / δ ⌉ + 1 for each graph of order n and with minimum degree δ > 0 .