Let $H_r(n,p)$ denote the maximum number of Hamiltonian cycles in an n-vertex r-graph with density $p\in (0,1)$. The expected number of Hamiltonian cycles in the random r-graph model $G_r(n,p)$ is $E(n,p)=p^n(n-1)!/2$ and in the random graph model $G_r(n,m)$ with $m=p\left(\begin{array}{c}n\\ r\end{array}\right)$ it is, in fact, slightly smaller than $E(n,p)$.

For graphs, $H_2(n,p)$ is proved to be only larger than $E(n,p)$ by a polynomial factor and it is an open problem whether a quasi-random graph with density p can be larger than $E(n,p)$ by a polynomial factor.

For hypergraphs (i.e. $r\ge 3$) the situation is drastically different. For all $r\ge 3$ it is proved that $H_r(n,p)$ is larger than $E(n,p)$ by an exponential factor and, moreover, there are quasi-random r-graphs with density p whose number of Hamiltonian cycles is larger than $E(n,p)$ by an exponential factor.

让 $H_r(n,p)$表示具有密度的n顶点r图中的哈密顿循环的最大数量$p\in (0,1)$. 随机r- graph 模型中哈密顿循环的预期数量$G_r(n,p)$ 是 $乙(n,p)=p^n(n-1)！/2$ 并在随机图模型中 $G_r(n,米)$ 和 $米=p\left(\begin{array}{c}n\\ r\end{array}\right)$ 事实上，它比 $乙(n,p)$.

对于图形， $H_2(n,p)$ 被证明只大于 $乙(n,p)$通过多项式因子，密度为p的拟随机图是否可以大于$乙(n,p)$ 由多项式因子。

对于超图（即 $r\ge 3$）情况大不相同。对所有人$r\ge 3$ 事实证明 $H_r(n,p)$ 大于 $乙(n,p)$通过指数因子，此外，存在密度为p的准随机r图，其哈密顿循环的数量大于$乙(n,p)$ 由指数因子。