THE DIAMETER OF RANDOM CAYLEY DIGRAPHS OF GIVEN DEGREE MANUEL E. LLADSER, PRIMO ˇ Z POTO ˇ CNIK, JANA ˇ SIAGIOV ´ A, JOZEF ˇ SIR ´ A ˇ N, AND MARK C. WILSON Abstract. We consider random Cayley digraphs of order n with uniformly distributed gener- ating sets of size k. Specifically, we are interested in the asymptotics of the probability that such a Cayley digraph has diameter two as n →∞ and k = f (n), focusing on the functions f (n)= ⌊cn⌋ and f (n)= ⌊n δ ⌋. In both instances we show that the probability converges to 1 as n →∞, for any fixed c ∈ (0, 1/2) and any fixed δ ∈ (1/2, 1), respectively. We obtain sharper results for Abelian groups. The proofs use detailed asymptotic analysis in several regimes of the combinatorial function a(n, k, t), equal to the number of ways of choosing a subset of size k from a set of size n while not choosing any of t preassigned disjoint pairs. 1. Introduction It is well known that almost all graphs and digraphs have diameter two [1]. This result has been generalized and strengthened in various directions, of which we shall be interested in restrictions to Cayley graphs and digraphs. In [5] it was proved that almost all Cayley digraphs have diameter two, and in [4] this was extended to Cayley graphs. The random model used in [5, 4] is the most straightforward one: in terms of Cayley digraphs for a given group G, one chooses a random generating set by choosing its elements among the non-identity elements of G independently and uniformly, each with probability 2 −n+1 where n is the order of G. Observe that such generating sets have size at least n/2 with probability at least 1/2, in which case a simple counting argument shows that the corresponding Cayley digraphs have diameter at most two. The less trivial part of [5] therefore concerns random Cayley digraphs in which the number of generators is at most half of the order of the group. This motivates a study of random Cayley digraphs in which the number of generators is re- stricted. In this case one cannot use the model of [5]. Instead, we let every generating set of the Cayley digraph of a fixed degree appear with equal probability. The fundamental question here is: For which functions f is it true that the diameter of a random Cayley digraph of an arbitrary group of order n and of degree f (n) is asymptotically almost surely equal to 2 as n tends to infinity? By the well known Moore bound for graphs or digraphs of diameter two we know that f has to increase at least as fast as √ n. A study of the behaviour of the problem for functions of the form f (n)= ⌊n δ ⌋ for the powers δ satisfying 1/2 ≤ δ< 1 is therefore natural in this context. However, even the case when f (n)= ⌊cn⌋ for a constant c seems not to have been investigated before and, as we shall see, leads to an interesting asymptotic analysis. The probability that a random Cayley digraph of (in- and out-) degree k on a group of order n has diameter 2 will be estimated in Section 3 in terms of a certain combinatorial function that depends on n, f (n), and a third parameter reflecting the class of groups considered. It turns out that in the case of Abelian groups the combinatorial function can be analyzed by elementary methods and yields the following results, proved in Section 4: • For any c such that 0 <c< 1/2, the probability of a random Cayley digraph on an Abelian group of order n and degree ⌊cn⌋ having diameter 2 is at least 1 − O(exp(−c 2 n/2)). • For any δ such that 1/2 <δ< 1, the probability of a random Cayley digraph on an Abelian group of order n and degree ⌊n δ ⌋ having diameter 2 is at least 1 − O(exp(−n 2δ−1 /2)). • For each µ(n) ∈ (0, 1] and ε> 0, the probability of a random Cayley digraph of degree k = ⌊  2n ln(n/µ(n))⌋ on an Abelian 2-group of order n having diameter 2 is at least 1 − µ(n) − ε for all sufficiently large n. • If f (n) is such that f (n)/ √ n ln n →∞ as n →∞, then the probability of a random Cayley digraph of degree f (n) on an Abelian group of order n having diameter 2 converges to 1. Date : April 12, 2011. 1