Random Structures & Algorithms
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Given a group G, the model 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@?R"+ we say that c is the threshold for diameter 2 for (G"k) if for any @e0 with high probability @C@?G(G"k,p) has diameter greater than 2 if p==(c+@e)lognn. In Christofides and Markstrom (in press) [5] we proved that if c is a threshold for diameter 2 for a family of groups (G"k) then c@?[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@?[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@?[1/4,4/3] is a threshold but a c@?(4/3,2] is a threshold if and only if it is of the form 4n/(3n-1) for some positive integer n.