Abstract
An assignment of positive integer weights to the edges of a simple graph G is called irregular, if the weighted degrees of the vertices are all different. The irregularity strength, s(G), is the maximal weight, minimized over all irregular assignments. In this study, we show that s(G) ≤ c1 n-δ, for graphs with maximum degree Δ ≤ n1-2 and minimum degree δ, and s(G) ≤ c2(log n)n-δ, for graphs with Δ > n1-2, where c1 and c2 are explicit constants. To prove the result, we are using a combination of deterministic and probabilistic techniques. © 2002 Wiley Periodicals, Inc. J Graph Theory 41: 120–137, 2002
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