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 $\{1,2,\dots ,k\}$-weighting of the edges of $G$ attributing distinct weighted degrees to all vertices, or equivalently the least $k$ enabling obtaining 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 $s(G)\le 3$. More generally, they also posed a similar question regarding the upper bound $s(G)\le K$ for any given constant $K$. We provide asymptotically tight solutions of these problems by proving that such optimal lower bound is of order $\frac{1}{K-1}n$ for every fixed integer $K\ge 3$.