# 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.

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## Funding

Research supported by NSF Award DMS-1953958.

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Przybyło, J., Wei, F. On the Asymptotic Confirmation of the Faudree–Lehel Conjecture for General Graphs. Combinatorica 43, 791–826 (2023). https://doi.org/10.1007/s00493-023-00036-5

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• DOI: https://doi.org/10.1007/s00493-023-00036-5