Enumerating sparse uniform hypergraphs with given degree sequence and forbidden edges

TL;DR

This paper provides an asymptotic enumeration formula for sparse uniform hypergraphs with a given degree sequence that avoid certain edges, and applies it to analyze properties like perfect matchings and Hamilton cycles.

Contribution

It introduces a new asymptotic enumeration method for hypergraphs with degree constraints and forbidden edges, extending previous combinatorial enumeration techniques.

Findings

Derived an enumeration formula for hypergraphs with given degree sequences and forbidden edges.

Obtained asymptotic counts for hypergraphs containing all edges of a specified set.

Estimated the expected number of perfect matchings and Hamilton cycles in random hypergraphs.

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.