Summary: 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 n2
have to be deleted from it to make it H-free. It is known
that in this case G contains at least f( , H)nh
H. We show that the largest possible function f( , H) is
polynomial in if and only if H is bipartite. This implies
that there is a one-sided error property tester for check-
ing H-freeness, whose query complexity is polynomial in
1/ , if and only if H is bipartite.
All graphs considered here are finite, undirected, and
have neither loops nor parallel edges.
Let P be a property of graphs, that is, a family of
graphs closed under graph isomorphism. A graph G with