On the Asymptotic Confirmation of the Faudree–Lehel Conjecture for General Graphs

Abstract

Given a simple graph G, the irregularity strength of G, denoted by s(G), is the least positive integer k such that there is a weight assignment on edges $f: E(G) \rightarrow \{1,2,\dots , k\}$ 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 $s(G) \ge n/d$ 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 $s(G) \le n/d + c$ 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 $s(G) \le n/{\delta }+ c$ for every graph G on n vertices with minimum degree ${\delta }\ge 1$ which does not contain an isolated edge; In this paper we confirm that the generalized Faudree–Lehel Conjecture holds for graphs with ${\delta }\ge n^\beta$ 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 ${\delta }\ge 1$ and without isolated edges, $s(G) \le \frac{n}{{\delta }}(1+\frac{c_1}{{\delta }^{\varepsilon }})+c_2$; We thereby extend in various aspects and strengthen a recent result of Przybyło, who showed that $s(G) \le \frac{n}{d}\big (1+ \frac{1}{\ln ^{\varepsilon /19}n} \big )=\frac{n}{d}(1+o(1))$ for d-regular graphs with $d\in [\ln ^{1+\varepsilon } n, n/\ln ^{\varepsilon }n]$. We also improve the earlier general upper bound: $s(G)< 6\frac{n}{{\delta }}+6$ of Kalkowski, Karoński and Pfender.