Bipartite Subgraphs and Quasi-randomness

Abstract

Abstract. We say that a family of graphs G = {Gn: n ≥ 1} is p-quasi-random, 0 < p < 1, if it shares typical properties of the random graph G(n, p); for a definition, see below. We denote by Q w � (p) the class of all graphs H for which e(Gn) ≥ (1 + and the number of not necessarily induced labeled copies of H in Gn is at o(1))p �n 2 most (1 + o(1))p e(H) n v(H) imply that G is p-quasi-random. In this note, we show that all complete bipartite graphs Ka,b, a, b ≥ 2, belong to Q w (p) for all 0 < p < 1. 1. Notation We start with fixing notation. For positive integers k, n and a real number x, we set [n] = {1,..., n} and (x)k = x(x − 1) × · · · × (x − k + 1). Given a graph G with vertex set V (G) and edge set E(G), v(G) stands for |V (G) | and e(G) for |E(G)|. Furthermore, for a subset X of V (G), G[X] denotes the subgraph induced by the vertices of X, and e(X) denotes the number of edges of G[X]. Given a vertex x ∈ V (G), NG(x) is the set of all vertices adjacent to x and, similarly, for a subset X of V (G), NG(X) denotes the set of all vertices adjacent to every vertex in X. Clearly, NG(X) = � x∈X NG(x). We also pu