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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∋uw(e)+w(u)≠∑e∋vw(e)+w(v) for every pair u,v of vertices of G. We prove that the total vertex irregularity strength of graphs with δ≥n0.5lnn is bounded from above by (2+o(1))nδ+4 . One of the cornerstones of the proof is a random ordering of the vertices generated by order statistics.