Short Proof of the Asymptotic Confirmation of the Faudree-Lehel Conjecture

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 (Formula present) for which each vertex weight (Formula present) is unique amongst all (Formula present). In 1987, Faudree and Lehel conjectured that there is a constant c such that (Formula present) for all d-regular graphs G on n vertices with d > 1, whereas it is trivial that (Formula present) In this short note we prove that the Faudree-Lehel Conjecture holds (Formula present) for any fixed ɛ > 0, with a small additive constant c = 28 for n large enough. Furthermore, we confirm the conjecture asymptotically by proving that for any fixed (Formula present) there is a constant C such that for all d-regular graphs (Formula present) extending and improving a recent result of Przybyłlo that (Formula present) and n is large enough.

Duke Scholars

Author Fan Wei Mathematics