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arXiv 0811.0949math.COmath.PR
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关于渗流和双层床猜想

On percolation and the bunkbed conjecture

Svante Linusson

备注：
13 pages, improved exposition thanks to anonymous referee. To appear in CPC
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摘要：

研究了乘积图$G\times K_2$上的边渗透问题.这里$G$是任何有限图，$K_2$由两个顶点$\{0,1\}$通过边连接组成。$G\times K_2$中的每条边以独立于其他边的概率$p$存在。双层床猜想指出，对于所有$G$和$p$，$(u,0)$与$(v,0)$在同一分量中的概率大于或等于对于每对顶点$u,v\in G$，$(u,0)$与$(v,1)$在同一分量中的概率。我们推广了这个猜想，并制定和证明了类似的声明随机有向图。的方法导致证明原来的猜想为特殊类的图$G$，特别是outerplanar图。

英文摘要：

We study a problem on edge percolation on product graphs $G\times K_2$. Here $G$ is any finite graph and $K_2$ consists of two vertices $\{0,1\}$ connected by an edge. Every edge in $G\times K_2$ is present with probability $p$ independent of other edges. The Bunkbed conjecture states that for all $G$ and $p$ the probability that $(u,0)$ is in the same component as $(v,0)$ is greater than or equal to the probability that $(u,0)$ is in the same component as $(v,1)$ for every pair of vertices $u,v\in G$. We generalize this conjecture and formulate and prove similar statements for randomly directed graphs. The methods lead to a proof of the original conjecture for special classes of graphs $G$, in particular outerplanar graphs.

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