2 results
Counting spanning subgraphs in dense hypergraphs
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- Journal:
- Combinatorics, Probability and Computing , First View
- Published online by Cambridge University Press:
- 30 May 2024, pp. 1-13
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We give a simple method to estimate the number of distinct copies of some classes of spanning subgraphs in hypergraphs with a high minimum degree. In particular, for each
$k\geq 2$ and 
$1\leq \ell \leq k-1$, we show that every 
$k$-graph on 
$n$ vertices with minimum codegree at leastcontains
\begin{equation*} \left \{\begin {array}{l@{\quad}l} \left (\dfrac {1}{2}+o(1)\right )n & \text { if }(k-\ell )\mid k,\\[5pt] \left (\dfrac {1}{\lceil \frac {k}{k-\ell }\rceil (k-\ell )}+o(1)\right )n & \text { if }(k-\ell )\nmid k, \end {array} \right . \end{equation*}
$\exp\!(n\log n-\Theta (n))$ Hamilton 
$\ell$-cycles as long as 
$(k-\ell )\mid n$. When 
$(k-\ell )\mid k$, this gives a simple proof of a result of Glock, Gould, Joos, Kühn, and Osthus, while when 
$(k-\ell )\nmid k$, this gives a weaker count than that given by Ferber, Hardiman, and Mond, or when 
$\ell \lt k/2$, by Ferber, Krivelevich, and Sudakov, but one that holds for an asymptotically optimal minimum codegree bound.
Spanning subdivisions in Dirac graphs
- Part of
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 20 October 2023, pp. 121-128
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We show that for every
$n\in \mathbb N$ and 
$\log n\le d\lt n$, if a graph 
$G$ has 
$N=\Theta (dn)$ vertices and minimum degree 
$(1+o(1))\frac{N}{2}$, then it contains a spanning subdivision of every 
$n$-vertex 
$d$-regular graph.
