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 $K_{r,s}$ on a subset of the vertices in $G$. We show that $G$ contains $H$ as a subgraph if and only if $β_{i,α}(H) \le β_{i,α}(G)$ for all $i \ge 0$ and $α\in \mathbb{Z}^n$. In fact, it suffices to consider only the first syzygy module. In particular, we prove that $β_{1,α}(H) \le β_{1,α}(G)$ for all $α\in \mathbb{Z}^n$ if and only if $G$ contains a subgraph that is isomorphic to either $H$ or a multipartite graph $K_{2,\dots,2,a,b}$.

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