r-图和拟随机r-图中的超期望哈密顿圈
Hamiltonian cycles above expectation in r-graphs and quasi-random r-graphs
摘要：
设$H_r(n,p)$表示密度为$p \in (0,1)$的$n$ -顶点$r$ -图中的最大哈密顿圈数.随机$r$图模型$G_r(n,p)$中的哈密顿圈的期望数目是$E(n,p)=p^n(n-1)!/2$，而在具有$m=p\binom{n}{r}$的随机图模型$G_r(n,m)$中，它实际上略小于$E(n,p)$ .对于图，证明了$H_2(n,p)$只比$E(n,p)$大一个多项式因子，而密度为$p$的拟随机图是否能比$E(n,p)$大一个多项式因子是一个公开问题.对于超图（即$r \ge 3$），情况完全不同。对所有的$r \ge 3$，证明了$H_r(n,p)$比$E(n,p)$大一个{ \em指数}因子，并且存在密度为$p$的拟随机$r$ -图，其哈密顿圈数比$E(n,p)$大一个指数因子.
英文摘要：
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 $m=p\binom{n}{r}$ 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 {\em 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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
