Szczegóły publikacji
Opis bibliograficzny
Short proof of the asymptotic confirmation of the Faudree-Lehel conjecture / Jakub PRZYBYŁO, Fan Wei // The Electronic Journal of Combinatorics [Dokument elektroniczny]. — Czasopismo elektroniczne ; ISSN 1077-8926. — 2023 — vol. 30 iss. 4 art. no. P4.27, s. 1-13. — Wymagania systemowe: Adobe Reader. — Bibliogr. s. 12-13, Abstr. — Publikacja dostępna online od: 2023-11-17
Autorzy (2)
- AGHPrzybyło Jakub
- Wei Fan
Dane bibliometryczne
| ID BaDAP | 150798 |
|---|---|
| Data dodania do BaDAP | 2024-01-03 |
| Tekst źródłowy | URL |
| DOI | 10.37236/11413 |
| Rok publikacji | 2023 |
| Typ publikacji | artykuł w czasopiśmie |
| Otwarty dostęp |  |
| Creative Commons | |
| Czasopismo/seria | The Electronic Journal of Combinatorics |
Abstract
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)→{1,2,…,k} for which each vertex weight fV(v):=∑u:{u,v}∈E(G)f({u,v}) is unique amongst all v∈V(G). In 1987, Faudree and Lehel conjectured that there is a constant c such that s(G)≤n/d+c for all d-regular graphs G on n vertices with d>1, whereas it is trivial that s(G)≥n/d. In this short note we prove that the Faudree-Lehel Conjecture holds when d≥n0.8+ϵ for any fixed ϵ>0, with a small additive constant c=28 for n large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed β∈(0,1/4) there is a constant C such that for all d-regular graphs G, s(G)≤nd(1+Cdβ)+28, extending and improving a recent result of Przybyło that s(G)≤nd(1+1lnϵ/19n) whenever d∈[ln1+ϵn,n/lnϵn] and n is large enough.
