KPKVB模型中的Hamilton圈与完美匹配
Hamilton cycles and perfect matchings in the KPKVB model
摘要：
本文研究了Krioukov等人提出的随机图模型中Hamilton圈和完美匹配的存在性。2010年在该模型中，节点在双曲平面内的圆盘内随机选择，如果两个节点之间至多有一定的双曲距离，则它们是连通的。先前已经表明，该模型具有与复杂网络相关的各种属性，包括幂律度分布，“短距离”和严格正的聚类系数。该模型使用三个参数指定：节点数$n$，我们认为它会趋于无穷大，以及$α, ν> 0$，我们认为它是常数。粗略地说，$α$控制度序列的幂律指数，而$ν$控制平均度。在这里，我们表明，对于每一个$α< 1/2$和$ν=ν(α)$足够小，该模型不包含一个完美的匹配与高概率，而对于每一个$α< 1/2$和$ν=ν(α)$足够大，该模型包含一个汉密尔顿循环与高概率。
英文摘要：
In this paper we consider the existence of Hamilton cycles and perfect matchings in a random graph model proposed by Krioukov et al.~in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, "short distances" and a strictly positive clustering coefficient. The model is specified using three parameters: the number of nodes $n$, which we think of as going to infinity, and $α, ν> 0$, which we think of as constant. Roughly speaking $α$ controls the power law exponent of the degree sequence and $ν$ the average degree. Here we show that for every $α< 1/2$ and $ν=ν(α)$ sufficiently small, the model does not contain a perfect matching with high probability, whereas for every $α< 1/2$ and $ν=ν(α)$ sufficiently large, the model contains a Hamilton cycle with high probability.
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