Betti numbers of subgraphs

Let G be a simple graph on n vertices. Let H be either the complete graph K-m or the complete bipartite graph K-r,K-s on a subset of the vertices in G. We show that G contains H as a subgraph if and only if beta(1,alpha)(H) <= beta(1,alpha) (G) for all i >= 0 and alpha is an element of Z(n). In fact, it suffices to consider only the first syzygy module. In particular, we prove that,31, (H) <,31, (G) for all it alpha is an element of Z(n) if and only if G contains a subgraph that is isomorphic to either H or a multipartite graph K-2,K-...,K-2,(a,b).