AI 中文总结
该研究将边染色随机图的彩虹哈密顿Berge圈结果推广到k≥3的随机k均匀超图，确定了对应的c和p阈值，且该阈值渐近紧。
AI 中文摘要
设H ~ H^k_c(n,p)是顶点集为[n]的边染色随机k均匀超图，其中每个边e ∈ C([n],k)以概率p独立出现，且从颜色集[c]中均匀独立地分配一个颜色。对于k=2，Ferber和Krivelevich（2016）证明，若c=(1+o(1))n且p=(log n + log log n + ω(n))/n，则高概率下边染色随机图H ~ H^2_c(n,p)包含彩虹哈密顿Berge圈。随后，Bal、Berkowitz、Devlin和Schacht（2021）确定了随机k均匀超图中（非彩虹）哈密顿Berge圈出现的阈值。本文将这些结果推广到所有整数k≥3，证明若c=(1+o(1))n且p=(k-1)!·(log n + log log n + ω(n))/n^{k-1}，则高概率下H ~ H^k_c(n,p)包含彩虹哈密顿Berge圈，且c和p的这两个条件都是渐近紧的。
英文摘要
Let $H \sim H^{k}_c(n,p)$ be an edge-colored random $k$-uniform hypergraph on the vertex set $[n]$, where each edge $e \in \binom{[n]}{k}$ is included independently with probability $p$ and is uniformly and independently assigned a color from the color set $[c]$. For $k = 2$, Ferber and Krivelevich (2016) established that if $c = (1+o(1))n$ and $p = (\log n + \log \log n + ω(n))/n$, then with high probability the edge-colored random graph $H \sim H^2_c(n,p)$ contains a rainbow Hamilton Berge cycle. Subsequently, Bal, Berkowitz, Devlin, and Schacht (2021) determined the threshold for the appearance of a (non-rainbow) Hamilton Berge cycle in random $k$-uniform hypergraphs. In this paper, we generalize the results to all integers $k \ge 3$. We prove that if $c = (1+o(1))n$ and $p = (k-1)! \frac{\log n + \log\log n + ω(n)}{n^{k-1}}$, then with high probability $H \sim H^{k}_c(n,p)$ contains a rainbow Hamilton Berge cycle. Furthermore, both conditions on $c$ and $p$ are asymptotically tight. \noindent\emph{Key words:} Rainbow subgraph, Hamiltonicity, Berge cycle, Random hypergraph.
