Counting and packing Hamilton $\ell$-cycles in dense hypergraphs

Abstract

We consider problems about packing and counting Hamilton $\ell$-cycles in hypergraphs of large minimum degree. Given a hypergraph $\mathcal H$, for a $d$-subset $A\subseteq V(\mathcal H)$, we denote by $d_{\mathcal H}(A)$ the number of distinct \emph{edges} $f\in E(\mathcal H)$ for which $A\subseteq f$, and set $\delta_d(\mathcal H)$ to be the minimum $d_{\mathcal H}(A)$ over all $A\subseteq V(\mathcal H)$ of size $d$. We show that if a $k$-uniform hypergraph on $n$ vertices $\mathcal H$ satisfies $\delta_{k-1}(\mathcal H)\geq \alpha n$ for some $\alpha>1/2$, then for every $\ell<k/2$ $\mathcal H$ contains $(1-o(1))^n\cdot n!\cdot \left(\frac{\alpha}{\ell!(k-2\ell)!}\right)^{\frac{n}{k-\ell}}$ Hamilton $\ell$-cycles. The exponent above is easily seen to be optimal. In addition, we show that if $\delta_{k-1}(\mathcal H)\geq \alpha n$ for $\alpha>1/2$, then $\mathcal H$ contains $f(\alpha)n$ edge-disjoint Hamilton $\ell$-cycles for an explicit function $f(\alpha)>0$. For the case where every $(k-1)$-tuple $X\subset V({\mathcal H})$ satisfies $d_{\mathcal H}(X)\in (\alpha\pm o(1))n$, we show that $\mathcal H$ contains edge-disjoint Haimlton $\ell$-cycles which cover all but $o\left(|E(\mathcal H)|\right)$ edges of $\mathcal H$. As a tool we prove the following result which might be of independent interest: For a bipartite graph $G$ with both parts of size $n$, with minimum degree at least $\delta n$, where $\delta>1/2$, and for $p=\omega(\log n/n)$ the following holds. If $G$ contains an $r$-factor for $r=\Theta(n)$, then by retaining edges of $G$ with probability $p$ independently at random, w.h.p the resulting graph contains a $(1-o(1))rp$-factor.