Product irregularity strength of certain graphs

Abstract

Consider a simple graph G with no isolated edges and at most one isolated vertex. A labeling w: E(G) → {1, 2, …, m} is called product - irregular, if all product degrees pd_G(v) = ∏ _{e ∋ v}w(e) are distinct. The goal is to obtain a product - irregular labeling that minimizes the maximal label. This minimal value is called the product irregularity strength and denoted ps(G). We give the exact values of ps(G) for several families of graphs, as complete bipartite graphs K_m, n, where 2 ≤ m ≤ n ≤ (m + 2) choose 2, some families of forests, including complete d-ary trees, and other graphs with δ(G) = 1.