发表机构
HIT – Holon Institute of Technology; University of Haifa（霍隆理工学院; 海法大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究随机顺序揭示环及其幂的顶点时最后一个孤立顶点出现的时间，通过逆转过程得到精确枚举公式，并证明普通环的极限为Rayleigh分布，其幂的极限为Weibull分布。
AI 中文摘要
设图的顶点以均匀随机顺序逐一揭示，最后隔离时间是指被揭示顶点诱导图包含一个孤立顶点的最后时刻。对于环C_n，记K_n为此时仍未揭示的顶点数。我们获得了K_n的精确有限n尾部公式，该公式以第二类Stirling数表示，并附带一个紧凑的二元生成函数。该枚举通过逆转过程得到。一个被揭示的孤立顶点对应于循环模式101，但尾部事件要求该模式在每一个更早的前缀中都不存在，而不仅仅是在最终集合中。这一前缀条件迫使最终的一块被长度至少为二的零间隙分隔，并迫使每个一块内部的揭示顺序是无峰的。由于大小为m的块有2^{m-1}种无峰顺序，对块大小求和便产生了Stirling数。我们还证明了K_n/sqrt(n) -> R，其中P(R>x)=e^{-x^2}，并且对于每个固定的d>=1，K_{n,d}/n^{1-1/(2d)} -> W_d，其中P(W_d>x)=e^{-x^{2d}}。均匀拉伸指数尾部界意味着所有固定正矩的收敛。因此，普通环具有Rayleigh极限，而其固定幂给出形状参数为2d的Weibull族。
英文摘要
Let the vertices of a graph be revealed one at a time in a uniformly random order, and let the last-isolation time be the last time at which the induced graph on the revealed vertices contains an isolated vertex. For the cycle C_n, write K_n for the number of vertices still unrevealed at this time. We obtain an exact finite-n tail formula for K_n in terms of Stirling numbers of the second kind, together with a compact bivariate generating function. The enumeration comes from reversing the process. An isolated revealed vertex then corresponds to the cyclic pattern 101, but the tail event requires this pattern to be absent from every earlier prefix, not merely from the final set. This prefix condition forces the final one-blocks to be separated by zero-gaps of length at least two and forces the reveal order inside each one-block to be peakless. Since a block of size m has 2^{m-1} peakless orders, summing over block sizes produces the Stirling numbers. We also prove K_n/sqrt(n) -> R, with P(R>x)=e^{-x^2}, and, for every fixed d>=1, K_{n,d}/n^{1-1/(2d)} -> W_d, with P(W_d>x)=e^{-x^{2d}}. Uniform stretched-exponential tail bounds imply convergence of all fixed positive moments. Thus the ordinary cycle has a Rayleigh limit, while its fixed powers give a Weibull family with shape parameter 2d.
Comments12 pages. Ancillary files contain computational verification code and recorded outputs
