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

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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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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

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