Betti numbers of subgraphs

Abstract

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