Irregularity strength of dense graphs

Let G be a simple graph of order n with no isolated vertices and no isolated edges. For a positive integer w, an assignment f on G is a function f : E(G) → {1, 2, . . . , w}. For a vertex v, f(v) is defined as the sum f(e) over all edges e of G incident with v. f is called irregular, if all f(v) are distinct. The smallest w for which there exists an irregular assignment on G is called the irregularity strength of G, and it is denoted by s(G). We show that if the minimum degree δ(G) ≥ 10n3/4 log n, then s(G) ≤ 48(n/δ)+6. For these δ, this improves the magnitude of the previous best upper bound of A. Frieze, R.J. Gould, M. Karoński, and F. Pfender by a log n factor. It also provides an affirmative answer to a question of J. Lehel, whether for every α ∈ (0, 1), there exists a constant c = c(α) such that s(G) ≤ c for every graph G of order n with minimum degree δ(G) ≥ (1 − α)n. Specializing the argument for d-regular graphs with d ≥ 104/3n2/3 log n, we prove that s(G) ≤ 48(n/d) + 6.