Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Counting and packing Hamilton $\ell$-cycles in dense hypergraphs   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.

16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
23
The categories of this paper are: solv-int
24
The categories of this paper are: dg-ga
0
The categories of this paper are: hep-ph
30
The categories of this paper are: q-alg
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
14
The categories of this paper are: math-ph
13
The categories of this paper are: quant-ph
32
The categories of this paper are: atom-ph
11
The categories of this paper are: hep-lat
27
The categories of this paper are: comp-gas
8
The categories of this paper are: hep-ex
29
The categories of this paper are: funct-an