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

TL;DR

This paper investigates the existence and packing of Hamilton ycles in dense hypergraphs with large minimum degree, providing asymptotically optimal counts and explicit bounds for edge-disjoint cycles.

Contribution

It establishes new thresholds and counts for Hamilton ycles in hypergraphs with large minimum degree, including optimal enumeration and packing results.

Findings

Hypergraphs with egree > 1/2 contain many Hamilton ycles.

Existence of isjoint Hamilton ycles proportional to egree.

Almost perfect edge coverage by Hamilton ycles 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.