A tight Hamilton cycle in a k-uniform hypergraph (k-graph) G is a cyclic ordering of the vertices of G such that every set of k consecutive vertices in the ordering forms an edge. Rödl, Ruciński and Szemerédi proved that for $k\ge 3$ , every k-graph on n vertices with minimum codegree at least $n/2+o(n)$ contains a tight Hamilton cycle. We show that the number of tight Hamilton cycles in such k-graphs is ${\exp(n\ln n-\Theta(n))}$ . As a corollary, we obtain a similar estimate on the number of Hamilton ${\ell}$ -cycles in such k-graphs for all ${\ell\in\{0,\ldots,k-1\}}$ , which makes progress on a question of Ferber, Krivelevich and Sudakov.

中文翻译：

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在狄拉克超图中计算汉密尔顿循环

一个紧的汉密尔顿循环ķ-均匀超图（ķ-图形）G是顶点的循环排序G这样每一组ķ排序中的连续顶点形成一条边。Rödl、Ruciński 和 Szemerédi 证明了$k\ge 3$， 每一个ķ- 上图n至少具有最小共度的顶点$n/2+o(n)$包含一个紧的汉密尔顿循环。我们表明，在这种情况下，紧汉密尔顿循环的数量ķ-图是${\exp(n\ln n-\Theta(n))}$. 作为推论，我们对汉密尔顿的数量获得了类似的估计${\ell}$-在这样的循环ķ- 所有人的图表${\ell\in\{0,\ldots,k-1\}}$，这在 Ferber、Krivelevich 和 Sudakov 的问题上取得了进展。