Simplicial Complexes of Graphs and Hypergraphs with a Bounded Covering Number

For $1 \le p \le n-1$, define $\Cov{n}{p}$ as the family of graphs on the vertex set $\{1,\ldots, n\}$ with a covering number of at most $p$ (equivalently, with an independence number of at least $n-p$). Since the underlying vertex set is fixed, we may identify each graph in $\Cov{n}{p}$ with its edge set. In particular, we may view $\Cov{n}{p}$ as a simplicial complex. For $i\ge -1$, we show that the rank of the $i$th homology group of $\Cov{n}{p}$ is a linear combination, with coefficients being polynomials in $n$, of the ranks of the $i$th homology groups of $\Cov{p+2}{p}, \ldots, \Cov{2p+1}{p}$. Our proof takes place in a more general setting where we consider complexes of hypergraphs. In addition, we show that the $(2p-1)$-skeleton of $\Cov{n}{p}$ is shellable, which implies that $\Cov{n}{p}$ is $(2p-2)$-connected. For $p \le 3$, we give a complete description of the homology groups of $\Cov{n}{p}$.