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arXiv 2102.12012math.CO
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稠密正则图随机子图的彩虹生成树

Rainbow spanning trees in random subgraphs of dense regular graphs

Peter Bradshaw

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11 pages, 1 figure
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摘要：

考虑边着色图的随机模型.在$n$顶点上的图$G$是固定的，并且通过使$G$的每条边以概率$p$保持独立来选择随机子图$G_p$。然后，从集合$[n-1]$中随机均匀地着色$G_p$的每个边缘。Frieze和McKay（Random Structures and Algorithms，1994）的一个结果表明，当$G = K_n$和$p = (2 + ε) \frac{\log n}{n}$对于某个常数$ε> 0$，则$G_p$几乎必然包含彩虹生成树。本文证明了：如果$G$是一个$d$ -正则$Ω(n)$ -边连通图，则当$p = (2 + ε) \frac {\log n}{d}$对某个常数$ε> 0$，$G_p$几乎必然包含一棵彩虹生成树.我们的主要工具是一个新的边缘替换方法的彩虹森林。

英文摘要：

We consider the following random model for edge-colored graphs. A graph $G$ on $n$ vertices is fixed, and a random subgraph $G_p$ is chosen by letting each edge of $G$ remain independently with probability $p$. Then, each edge of $G_p$ is colored uniformly at random from the set $[n-1]$. A result of Frieze and McKay (Random Structures and Algorithms, 1994) implies that when $G = K_n$ and $p = (2 + ε) \frac{\log n}{n}$ for some constant $ε> 0$, then $G_p$ almost surely contains a rainbow spanning tree. In this paper, we show that if $G$ is a $d$-regular $Ω(n)$-edge-connected graph, then when $p = (2 + ε) \frac {\log n}{d}$ for some constant $ε> 0$, $G_p$ almost surely contains a rainbow spanning tree. Our main tool is a new edge-replacement method for rainbow forests.

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。

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