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Total Vertex Irregularity Strength of Dense Graphs
Author(s) -
Majerski P.,
Przybyło J.
Publication year - 2014
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.21748
Subject(s) - combinatorics , mathematics , vertex (graph theory) , bounded function , graph , weighting , neighbourhood (mathematics) , discrete mathematics , medicine , mathematical analysis , radiology
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.