随机边序中的Hamilton增路
Hamiltonian increasing paths in random edge orderings
摘要：
如果完全图$K_n$的边是全序的，则边是升序的简单路称为增路。几十年来，最长递增路径的最坏情况长度一直是一个悬而未决的问题，其渐近界介于$\sqrt{n}$（Graham and Kleitman，1973）和$n/2$（Calderbank，Chung，and Sturtevant，1984）之间。我们考虑的平均情况下，当顺序是随机选择均匀。我们发现令人惊讶的结果，在随机设置中，一个最大可能长度的$n-1$的增加路径存在的概率至少约为$1/e$。我们还证明了概率$1-o(1)$，有一个增加的路径的长度至少为$0.85n$，这表明这种哈密顿（或近哈密顿）现象可以渐近几乎肯定。
英文摘要：
If the edges of the complete graph $K_n$ are totally ordered, a simple path whose edges are in ascending order is called increasing. The worst-case length of the longest increasing path has remained an open problem for several decades, with asymptotic bounds between $\sqrt{n}$ (Graham and Kleitman, 1973) and $n/2$ (Calderbank, Chung, and Sturtevant, 1984). We consider the average case, when the ordering is chosen uniformly at random. We discover the surprising result that in the random setting, an increasing path of the maximum possible length of $n-1$ exists with probability at least about $1/e$. We also prove that with probability $1-o(1)$, there is an increasing path of length at least $0.85n$, suggesting that this Hamiltonian (or near-Hamiltonian) phenomenon may hold asymptotically almost surely.
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