Irregularity strength of regular graphs

Abstract

Let G be a simple graph with no isolated edges and at most one isolated vertex. For a positive integer w, a w-weighting of G is a map f : E(G) → {1, 2, . . . , w}. An irregularity strength of G, s(G), is the smallest w such that there is a w-weighting of G for which ∑e:u∈e f(e) ≠ ∑e:v∈e f(e) for all pairs of different vertices u, v ∈ V (G). A conjecture by Faudree and Lehel says that there is a constant c such that s(G) ≤ n/d + c for each d-regular graph G, d ≥ 2. We show that s(G) < 16 n/d + 6. Consequently, we improve the results by Frieze, Gould, Karoński and Pfender (in some cases by a log n factor) in this area, as well as the recent result by Cuckler and Lazebnik.