Asymptotic confirmation of the Faudree-Lehel Conjecture on irregularity strength for all but extreme degrees

TL;DR

This paper proves that the irregularity strength of regular graphs asymptotically matches the conjectured bound for most degrees, except for extremely small or large degrees, advancing understanding of graph irregularity measures.

Contribution

It establishes that the Faudree-Lehel conjecture holds asymptotically for a wide range of degrees in regular graphs, except for extreme cases.

Findings

Proves $s(G) o rac{n}{d}$ asymptotically for most degrees.

Shows the conjecture holds when $d$ is between logarithmic powers of $n$.

Provides bounds for irregularity strength for large $n$ and various degree ranges.

Abstract

The irregularity strength of a graph $G$, $s(G)$, is the least $k$ admitting a $\{1,2,\ldots,k\}$-weighting of the edges of $G$ assuring distinct weighted degrees of all vertices, or equivalently the least possible maximal edge multiplicity in an irregular multigraph obtained of $G$ via multiplying some of its edges. The most well-known open problem concerning this graph invariant is the conjecture posed in 1987 by Faudree and Lehel that there exists a constant $C$ such that $s(G)\leq \frac{n}{d}+C$ for each $d$-regular graph $G$ with $n$ vertices and $d\geq 2$ (while a straightforward counting argument yields $s(G)\geq \frac{n+d-1}{d}$). The best known results towards this imply that $s(G)\leq 6\lceil\frac{n}{d}\rceil$ for every $d$-regular graph $G$ with $n$ vertices and $d\geq 2$, while $s(G)\leq (4+o(1))\frac{n}{d}+4$ if $d\geq n^{0.5}\ln n$. We show that the conjecture of Faudree and Lehel holds asymptotically in the cases when $d$ is neither very small nor very close to $n$. We in particular prove that for large enough $n$ and $d\in [\ln^8n,\frac{n}{\ln^3 n}]$, $s(G)\leq \frac{n}{d}(1+\frac{8}{\ln n})$, and thereby we show that $s(G) = \frac{n}{d}(1+o(1))$ then. We moreover prove the latter to hold already when $d\in [\ln^{1+\varepsilon}n,\frac{n}{\ln^\varepsilon n}]$ where $\varepsilon$ is an arbitrary positive constant.