Total Vertex Irregularity Strength of Dense Graphs

Consider a graph G = ( V , E ) of minimum degree δ and order n . Its total vertex irregularity strength is the smallest integer k for which one can find a weighting w : E ∪ V → { 1 , 2 , ... , k } such that∑ e ∋ u w ( e ) + w ( u ) ≠ ∑ e ∋ v w ( e ) + w ( v )for every pair u , v of vertices of G . We prove that the total vertex irregularity strength of graphs with δ ≥ n 0.5 ln n is bounded from above by( 2 + o ( 1 ) ) n δ + 4 . One of the cornerstones of the proof is a random ordering of the vertices generated by order statistics.