On the asymptotic confirmation of the Faudree-Lehel Conjecture for general graphs

Abstract

Given a simple graph $G$, the {\it irregularity strength} of $G$, denoted by $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) \to \{1,2,\dots, k\}$ attributing distinct weighted degrees: $\tilde f(v):= \sum_{u: \{u,v\}\in E(G)} f(\{u,v\})$ to all vertices $v\in V(G)$. It is straightforward that $s(G) \geq n/d$ for every $d$-regular graph $G$ on $n$ vertices with $d>1$. In 1987, Faudree and Lehel conjectured in turn that there is an absolute constant $c$ such that $s(G) \leq n/d + c$ for all such graphs. Even though the conjecture has remained open in almost all relevant cases, it is more generally believed that there exists a universal constant $c$ such that $s(G) \leq n/\delta + c$ for every graph $G$ on $n$ vertices with minimum degree $\delta \geq 1$ which does not contain an isolated edge. In this paper we confirm that the generalized Faudree-Lehel Conjecture holds for graphs with $\delta\geq n^\beta$ where $\beta$ is any fixed constant larger than $0.8$. Furthermore, we confirm that the conjecture holds in general asymptotically. That is we prove that for any $\varepsilon\in(0,0.25)$ there exist absolute constants $c_1, c_2$ such that for all graphs $G$ on $n$ vertices with minimum degree %at least $\delta\geq 1$ and without isolated edges, $s(G) \leq \frac{n}{\delta}(1+\frac{c_1}{\delta^{\varepsilon}})+c_2$, thus extending in various aspects and strengthening a recent result of Przyby{\l}o, who showed that $s(G) \leq \frac{n}{d}(1+ \frac{1}{\ln^{\epsilon/19}n})=\frac{n}{d}(1+o(1))$ for $d$-regular graphs with $d\in [\ln^{1+\epsilon} n, n/\ln^{\epsilon}n]$, and improving an earlier general upper bound: $s(G)< 6\frac{n}{\delta}+6$ of Kalkowski, Karo\'nski and Pfender.