Bounding the distant irregularity strength of graphs via a non-uniformly biased random weight assignment

Abstract

Given an edge k-weighting ω:E→[k] of a graph G=(V,E), the weighted degree of a vertex v∈V is the sum of its incident weights. The least k for which there exists an edge k-weighting such that the resulting weighted degrees of the vertices at distance at most r in G are distinct is called the r-distant irregularity strength, and denoted sr(G). This concept links the well-known 1–2–3 Conjecture, corresponding to s1(G), with the irregularity strength of graphs, s(G), which coincides with sr(G) for every r at least the diameter of G. It is believed that for every r≥2, sr(G)≤(1+o(1))Δr−1, where Δ is the maximum degree of G, while it is known that sr(G)≤6Δr−1 in general and sr(G)≤(4+o(1))Δr−1 for graphs with minimum degree δ at least log8Δ. We apply the probabilistic method in order to improve these results and show that graphs with δ≫lnΔ satisfy sr(G)≤(e+o(1))Δr−1 as Δ→∞.