{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,4]],"date-time":"2025-11-04T16:19:50Z","timestamp":1762273190331,"version":"3.41.2"},"reference-count":0,"publisher":"The Electronic Journal of Combinatorics","issue":"4","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Electron. J. Combin."],"abstract":"<jats:p>Given a simple graph $G$, the 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 $v\\in V(G)$. 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 $d&gt;1$, 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 $\\epsilon &gt;0$, with a small additive constant $c=28$ for $n$ large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed $\\beta\\in(0,1\/4)$ 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 Przyby\u0142o that $s(G) \\leq \\frac{n}{d}(1+ \\frac{1}{\\ln^{\\epsilon\/19}n})$ whenever $d\\in [\\ln^{1+\\epsilon} n, n\/\\ln^{\\epsilon}n]$ and $n$ is large enough.<\/jats:p>","DOI":"10.37236\/11413","type":"journal-article","created":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T15:39:49Z","timestamp":1700149189000},"source":"Crossref","is-referenced-by-count":3,"title":["Short Proof of the Asymptotic Confirmation of the Faudree-Lehel Conjecture"],"prefix":"10.37236","volume":"30","author":[{"given":"Jakub","family":"Przyby\u0142o","sequence":"first","affiliation":[]},{"given":"Fan","family":"Wei","sequence":"additional","affiliation":[]}],"member":"23455","published-online":{"date-parts":[[2023,11,17]]},"container-title":["The Electronic Journal of Combinatorics"],"original-title":[],"link":[{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i4p27\/pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/download\/v30i4p27\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,11,16]],"date-time":"2023-11-16T15:39:50Z","timestamp":1700149190000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.combinatorics.org\/ojs\/index.php\/eljc\/article\/view\/v30i4p27"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,11,17]]},"references-count":0,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2023,10,6]]}},"URL":"https:\/\/doi.org\/10.37236\/11413","relation":{},"ISSN":["1077-8926"],"issn-type":[{"type":"electronic","value":"1077-8926"}],"subject":[],"published":{"date-parts":[[2023,11,17]]},"article-number":"P4.27"}}