The irregularity strength of dense graphs — on asymptotically optimal solutions of problems of Faudree, Jacobson, Kinch and Lehel

Abstract

The irregularity strength of a graph G, s(G), is the least k such that there exists a {1,2,…,k}-weighting of the edges of G attributing distinct weighted degrees to all vertices, or equivalently the least k allowing the creation of a multigraph with nonrecurring degrees by blowing each edge e of G to at most k copies of e. In 1991 Faudree, Jacobson, Kinch and Lehel asked for the optimal lower bound for the minimum degree of a graph G of order n which implies that s(G)≤3. More generally, they also posed a similar question regarding the upper bound s(G)≤K for any given constant K. We provide asymptotically tight solutions of these problems by proving that such optimal lower bound is of order 1K−1n for every fixed integer K≥3.