arXiv 1202.66732012-02-29math.CO

随机Cayley图直径为2的阈值范围

The range of thresholds for diameter 2 in random Cayley graphs

Demetres Christofides, Klas Markström

备注：: 14 pages

摘要：

给定群G，模型\mathcal{G}（G，p）表示G的所有Cayley图的概率空间，其中生成集的每个元素以概率p独立地随机选择。给定群族（G_k）和c \in \mathbb{R}_+，我们说c是（G_k）的直径为2的阈值，如果对于任何具有高概率Γ \varepsilon > 0的\in \mathcal{G}（G_k，p）如果是\leqslant \sqrt{(c - \eps)\frac{\log{n}}{n}}，则直径大于2，如果是\geqslant \sqrt{(c + \eps)\frac{\log{n}}{n}}，则直径至多为2。在[5]中，我们证明了如果c是群族（G_k）的直径为2的阈值，则c \in [1/4，2]，并给出了阈值分别为1/4和2的两个群族.本文研究了是否每一个c \in [1/4，2]都是某群族直径为2的阈值的问题。令人惊讶的是，这个问题的答案是否定的。我们证明了每个c \in [1/4，4/3]是一个阈值，但c \in（4/3，2]是一个阈值当且仅当它是4 n/（3 n-1）的形式，其中n是正整数.

英文摘要：

Given a group G, the model \mathcal{G}(G,p) denotes the probability space of all Cayley graphs of G where each element of the generating set is chosen independently at random with probability p. Given a family of groups (G_k) and a c \in \mathbb{R}_+ we say that c is the threshold for diameter 2 for (G_k) if for any \varepsilon > 0 with high probability Γ\in \mathcal{G}(G_k,p) has diameter greater than 2 if p \leqslant \sqrt{(c - \eps)\frac{\log{n}}{n}} and diameter at most 2 if p \geqslant \sqrt{(c + \eps)\frac{\log{n}}{n}}. In [5] we proved that if c is a threshold for diameter 2 for a family of groups (G_k) then c \in [1/4,2] and provided two families of groups with thresholds 1/4 and 2 respectively. In this paper we study the question of whether every c \in [1/4,2] is the threshold for diameter 2 for some family of groups. Rather surprisingly it turns out that the answer to this question is negative. We show that every c \in [1/4,4/3] is a threshold but a c \in (4/3,2] is a threshold if and only if it is of the form 4n/(3n-1) for some positive integer n.

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