Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges

Abstract

For $n\geq 3$ and $r=r(n) \geq 3$, let $\boldsymbol{k} =\boldsymbol{k}(n)=(k_1, \ldots, k_n)$ be a sequence of non-negative integers with sum $M(\boldsymbol{k})=\sum_{j=1}^{n} k_j$. We assume that $M(\boldsymbol{k})$ is divisible by $r$ for infinitely many values of $n$, and restrict our attention to these values. Let $X=X(n)$ be a simple $r$-uniform hypergraph on the vertex set $V=\{v_1,v_2, \ldots, v_n\}$ with $t$ edges and maximum degree $x_{\max}$. We denote by $\mathcal{H}_r(\boldsymbol{k})$ the set of all simple $r$-uniform hypergraphs on the vertex set $V$ with degree sequence $\boldsymbol{k}$, and let $\mathcal{H}_r(\boldsymbol{k},X)$ be the set of all hypergraphs in $\mathcal{H}_r(\boldsymbol{k})$ which contain no edge of $X$. We give an asymptotic enumeration formula for the size of $\mathcal{H}_r(\boldsymbol{k},X)$. This formula holds when $r^4 k_{\max}^3=o(M(\boldsymbol{k}))$, $t\, k_{\max}^{3}\, =o(M(\boldsymbol{k})^2)$ and $r\,t\,k_{\max}^4 = o(M(\boldsymbol{k})^3)$. Our proof involves the switching method. As a corollary, we obtain an asymptotic formula for the number of hypergraphs in $\mathcal{H}_r(\boldsymbol{k})$ which contain every edge of $X$. We apply this result to find asymptotic expressions for the expected number of perfect matchings and loose Hamilton cycles in a random hypergraph in $\mathcal{H}_r(\boldsymbol{k})$ in the regular case.