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- W3201939494 abstract "Given a simple graph $G$, the {it irregularity strength} of $G$, denoted $s(G)$, is the least positive integer $k$ such that there is a weight assignment on edges $f: E(G) to {1,2,dots, k}$ for which each vertex weight $f^V(v):= sum_{u: {u,v}in E(G)} f({u,v})$ is unique amongst all $vin V(G)$. In 1987, Faudree and Lehel conjectured that there is a constant $c$ such that $s(G) leq n/d + c$ for all $d$-regular graphs $G$ on $n$ vertices with $d>1$, whereas it is trivial that $s(G) geq n/d$. In this short note we prove that the Faudree-Lehel Conjecture holds when $d geq n^{0.8+epsilon}$ for any fixed $epsilon >0$, with a small additive constant $c=28$ for $d$ large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed $betain(0,1/4)$ there is a constant $C$ such that for all $d$-regular graphs $G$, $s(G) leq frac{n}{d}(1+frac{C}{d^{beta}})+28$, extending and improving a recent result of Przybylo that $s(G) leq frac{n}{d}(1+ frac{1}{ln^{epsilon/19}n})$ whenever $din [ln^{1+epsilon} n, n/ln^{epsilon}n]$ and $d$ is large enough." @default.
- W3201939494 created "2021-10-11" @default.
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- W3201939494 date "2021-09-27" @default.
- W3201939494 modified "2023-09-25" @default.
- W3201939494 title "Short proof of the asymptotic confirmation of the Faudree-Lehel Conjecture." @default.
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