Testing subgraphs in large graphs

Let H be a fixed graph with h vertices, let G be a graph on n vertices, and suppose that at least ϵn2 edges have to be deleted from it to make it H‐free. It is known that in this case G contains at least f(ϵ, H)nh copies of H. We show that the largest possible function f(ϵ, H) is polynomial in ϵ if and only if H is bipartite. This implies that there is a one‐sided error property tester for checking H‐freeness, whose query complexity is polynomial in 1/ϵ, if and only if H is bipartite. © 2002 Wiley Periodicals, Inc. Random Struct. Alg., 21: 359–370, 2002