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- W4298392506 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 $dgeq 2$ (while a straightforward counting argument yields $s(G)geq frac{n+d-1}{d}$). The best known results towards this imply that $s(G)leq 6lceilfrac{n}{d}rceil$ for every $d$-regular graph $G$ with $n$ vertices and $dgeq 2$, while $s(G)leq (4+o(1))frac{n}{d}+4$ if $dgeq n^{0.5}ln n$. 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 $din [ln^8n,frac{n}{ln^3 n}]$, $s(G)leq frac{n}{d}(1+frac{8}{ln n})$, and thereby we show that $s(G) = frac{n}{d}(1+o(1))$ then. We moreover prove the latter to hold already when $din [ln^{1+varepsilon}n,frac{n}{ln^varepsilon n}]$ where $varepsilon$ is an arbitrary positive constant." @default.
- W4298392506 created "2022-10-02" @default.
- W4298392506 creator A5009715410 @default.
- W4298392506 date "2019-12-17" @default.
- W4298392506 modified "2023-10-17" @default.
- W4298392506 title "Asymptotic confirmation of the Faudree-Lehel Conjecture on irregularity strength for all but extreme degrees" @default.
- W4298392506 doi "https://doi.org/10.48550/arxiv.1912.07858" @default.
- W4298392506 hasPublicationYear "2019" @default.
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