Asymptotic confirmation of the Faudree–Lehel conjecture on irregularity strength for all but extreme degrees

The irregularity strength of a graph G, s ( G ) , is the least k admitting { 1 2 … } -weighting edges assuring distinct weighted degrees all vertices, or equivalently possible maximal edge multiplicity in an irregular multigraph obtained via multiplying some its edges. most well-known open problem concerning this invariant conjecture posed 1987 by Faudree and Lehel that there exists constant C such ≤ n d + for each d-regular with vertices ≥ (while straightforward counting argument yields − ). best known results towards imply 6 ⌈ ⌉ every 2, while 4 o if 0.5 ln n. We show holds asymptotically cases when neither very small nor close to particular prove large enough ∈ [ 8 3 ] thereby we = then. moreover latter hold already ε where arbitrary positive constant.