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

TL;DR

This paper determines asymptotically optimal bounds on the minimum degree of dense graphs to ensure their irregularity strength is at most a given constant, solving longstanding questions in graph labeling theory.

Contribution

It provides asymptotically tight bounds on the minimum degree needed for dense graphs to have irregularity strength at most K, for any fixed K ≥ 3.

Findings

Optimal lower bounds are of order (1/(K-1))n for fixed K ≥ 3.

The results solve longstanding open problems from Faudree et al. (1991).

The bounds are asymptotically tight for dense graphs.

Abstract

The irregularity strength of a graph $G$, $s(G)$, is the least $k$ such that there exists a $\{1,2,\ldots,k\}$-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 $s(G)\leq 3$. More generally, they also posed a similar question regarding the upper bound $s(G)\leq K$ 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 $K\geq 3$.