Random Cayley digraphs of diameter 2 and given degree

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Introduction
It is well known that almost all graphs and digraphs have diameter two [Bol79].This result has been generalized and strengthened in various directions, of which we shall be interested in restrictions to Cayley graphs and digraphs.
In [ML97] it was proved that almost all Cayley digraphs have diameter two, and in [MH98] this was extended to Cayley graphs.The random model used in [ML97,MH98] is the most straightforward one: in terms of Cayley digraphs for a given group G of order n, one chooses a random generating set by choosing its elements among the non-identity elements of G independently and uniformly, each with probability 1/2.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 [ML97] 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 restricted.In this case one cannot use the model of [ML97].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 the asymptotic analysis of which yields the following main results, proved in Section 4: • For every c such that 0 < c < 1/2, the probability of a random Cayley digraph of degree cn on a given group of order n having diameter 2 is at least 1 − O(exp(−c 2 n/2)).
• For every δ such that 1/2 < δ < 1, the probability of a random Cayley digraph of degree n δ on a given group of order n having diameter 2 is at least • There is a constant c 1 such that for every function µ defined on positive integers, with µ(n) → ∞ as n → ∞, the probability of a random Cayley digraph of degree 2n ln(c 1 nµ(n)) on a group of order n having diameter 2 tends to 1 as n → ∞.
• There is a positive constant c 2 < 1 such that for every ε with 0 < ε < c 2 the probability of a random Cayley digraph of degree 2n ln(c 2 /ε) on an Abelian group of order n having diameter 2 has limes superior not exceeding 1 − ε.

The model
Throughout, let G be a finite group of order n and let k be a positive integer not exceeding n − 2. The set of non-trivial elements of G will be denoted by G * .For a set A and an integer r, the symbol A r stands for the set of all subsets of A of size r.
For S ∈ G * k , the Cayley digraph on G relative to S, denoted by Cay(G, S), is the k-valent digraph with vertex set G and arc set {(g, gs) : g ∈ G, s ∈ S}.The distance ∂ S (g, h) from the vertex g to the vertex h in Cay(G, S) is the length of the shortest directed path from g to h in Cay(G, S).The diameter diam(Cay(G, S)) is the smallest integer d such that for every ordered pair (g, h) the distance from g to h is at most d.
We are now ready to introduce our model for random Cayley digraphs of a given valence.Let P(G, k) be the probability space (B, 2 B , Pr) where B = G * k , 2 B is the power set of B, and Pr is the uniformly distributed probability measure on B.
More generally, for every subset L ⊆ G * of size , the probability that a random set S ∈ B contains L as a subset is given by Before proceeding, note that one could think of a seemingly easier model, where each element of G * is chosen to be a member of S independently and with probability k/(n − 1).This model may well be equivalent to our model and it may well be possible to prove their equivalence in a way similar to arguments about the equivalence of the two standard models of random graphs (cf.Chapter 2 of [Bol79]).In view of the length of the arguments of Chapter 2 of [Bol79], however, research into a possible equivalence of the two models should be the object of a separate article, which is the reason why we chose not to pursue this direction.
Let Diam : B → R be the random variable on the space P(G, k) defined by letting, for every The main goal of this article is to derive bounds on the probability of the event Diam(S) > 2 and study the asymptotic behavior of the bounds.Since Cayley digraphs are vertex-transitive, the diameter of Cay(G, S) coincides with the maximum value of ∂ S (1, y) over all y ∈ G * .Clearly, ∂ S (1, y) ≤ 2 if and only if y ∈ S, or there exists x ∈ S such that (1, x, y) is a directed path from 1 to y of length 2. The latter is equivalent to requiring that {x, x −1 y} ⊆ S; in particular, the events in the following definition play an important role in the analysis.
If S is an arbitrary element of G * k and Diam(S) ≤ 2 then, for each y ∈ G * , y ∈ S or S ∈ X(y).Hence Pr(Diam ≤ 2) ≤ min y∈G * Pr(y ∈ S or S ∈ X(y)).In particular, if Pr(X(y) | y / ∈ S) denotes the conditional probability of the complement X(y) of X(y) given that y / ∈ S, we have: where the last identity is a direct consequence of (1).On the other hand, if Diam(S) > 2 then there exists y ∈ G * such that y / ∈ S and S / ∈ X(y).This results in These inequalities immediately lead to the following result: where Notice that the upper and lower bounds in (3) differ by a linear factor of order n.These bounds will be used in Section 4 for asymptotic estimates on the probability of the event [Diam > 2].

Probability estimates
We begin with an auxiliary result on groups that we will need later.If G is a group and y ∈ G, we say that z ∈ G is a square root of y if z 2 = y.Lemma 3.1 Let G be a finite group.If the order of G is odd, then every y ∈ G has a unique square root in G.If the order of G is even, then there exists at least one element y ∈ G with no square root in G.
Proof: Consider the mapping s : G → G, x → x 2 .Suppose first that |G| = 2m + 1 is odd; in particular, x 2m+1 = 1, for all x ∈ G, due to Lagrange's theorem.Hence, for every x, y ∈ G, we see that x 2 = y 2 implies that x = x 2m+2 = y 2m+2 = y.So s is a bijection and therefore each y ∈ G has a unique square root in G if the group has odd order.On the other hand, if |G| is even, G has a non-trivial involution x.Since s(x) = 1 = s(1), s is not a bijection.Since G is finite, it follows that s is not surjection, and so there is an element y ∈ G with no square root in G. 2 The basis for our estimates on the probability Pr(Diam > 2) is provided by a bound on M which we state and prove next.
Proposition 3.2 For each group G of order n ≥ 3 and every k such that 1 ≤ k ≤ n − 2 we have with equality if G is Abelian.By Lemma 3.1 we know that if G is odd, then for every y ∈ G * there exists a unique x = x y ∈ G such that x 2 = y.For each y ∈ G * we let W y = G * y if |G| is even, and W y = G * \{y, x y } if |G| is odd.It can be checked that for every fixed y ∈ G * the mapping x → γ y (x) given by γ y (x) = x −1 y is a permutation of this newly introduced set W y .Notice that the size of W y is always even.Further, observe that for all y ∈ G * we have S ∈ X (y) if and only if S ⊂ W y , |S| = k, and S ∩ γ y (S) = ∅.Letting w(γ y ) = |{S ⊂ W y : |S| = k, S ∩ γ y (S) = ∅}|, this observation combined with (6) gives To estimate w(γ y ) we will extend the definition of w to arbitrary permutations by letting, for an arbitrary permutation α y of the set W y , Our strategy will be to show that for every y ∈ G * there exists an involution β y on the set W y such that w(γ y ) ≤ w(β y ) = t k 2 k .We will construct β y by defining it on every orbit O ⊂ W y of γ y successively.Since the construction will work for all y ∈ G * we will suppress subscripts on permutations and let W y = W , γ y = γ and β y = β in the description of the construction.Suppose first that |O| is even.The restriction of γ on O is a cyclic permutation; let O 1 and O 2 be the two orbits of γ 2 on O. Define the restriction of β on O by letting The case of orbits of odd size requires more attention.Since |W | is even, γ has an even number of orbits of odd size on W .Let us partition the set of odd-sized orbits of γ into ordered pairs and let (O 1 , O 2 ) be such a pair.Then, γ induces cyclic permutations, say, (u 1 , . . ., u 2r+1 ) and (v 1 , . . ., v 2s+1 ), of O 1 and O 2 , respectively, for some r, s ≥ 0. Define the restriction of We construct a new set T o out of T as follows.If u 1 , u 3 , v 1 , v 3 ∈ T , we take the smallest i and j such that both u 2i+1 and v 2j+1 lie outside T ; the condition T ∩ γ(T ) = ∅ implies that such i and j exist and 2 ≤ i ≤ r − 1 and 2 ≤ j ≤ s − 1.In this situation we let , where i is defined as before.In the case when v 1 , v 3 , u 1 ∈ T but u 3 / ∈ T we proceed symmetrically, swapping u and v (with appropriate subscripts).If In all these cases it is easy to check that T o ∩ β(T o ) = ∅.In all the remaining cases we let Finally, let S be a subset of W such that |S| = k and S ∩ γ(S) = ∅.We define S by letting S ∩ O = S ∩ O for every even-sized orbit O of γ, and odd-sized orbits of γ from our fixed pairing of such orbits, as introduced above.
The key point to observe is that the assignment S → S is injective.Moreover, the facts about the sets T and T o listed in the previous two paragraphs imply that S ∩ β(S ) = ∅.
The above arguments prove that w(γ y ) ≤ w(β y ) for all y ∈ G * .Further, since β y is a product of t = (n − 2)/2 cycles of length 2, the quantity w(β y ) defined in (8) is equal to t k 2 k since a set S ⊂ W y with |S| = k such that S ∩ β y (S) = ∅ arises precisely by choosing k cycles of β y length 2 out of t such cycles and choosing one of the two elements in each chosen 2-cycle.These facts combined with (7) and (6) imply the bound in our Proposition for general groups.
If G is Abelian, choose y ∈ G * arbitrarily if G has odd order, and let y ∈ G * be an element with no square root if the order of G is even.Then the permutation γ y of W y is an involution with no fixed point and hence β y = γ y in this case, implying equality for Abelian groups. 2 In combination with Proposition 2.2 this gives the following consequence.
for general groups, and for Abelian groups. 2

Asymptotic analysis
In the context of Proposition 3.3 with t = (n − 2)/2 we write where For the asymptotic analysis of the behaviour of binomial coefficients appearing in b(t, k) we use Stirling's approximation.To state the result of the corresponding routine calculations in a concise form, for , and Then, writing k = λt, routine calculation with the help of Stirling's approximation yields the terms R, P and C represent the exponential rate, the leading coefficient, and the correction term.The exponential rate R(λ) is easily seen to be negative for 0 < λ < 1. Furthermore C(t, λ) tends to 1 for each fixed λ ∈ (0, 1) as t → ∞.Note also that Our first result about the behaviour of Pr(Diam ≤ 2) in the case of general groups deals with the situation where k ≈ cn for some c < 1/2.In our asymptotic calculations for n → ∞ we may then replace λ = k/t for t = (n − 2)/2 with the value 2c.Combining now (9), (10) and (11).with the lower bound of Proposition 3.3 we arrive at the following result: Theorem 4.1 For each c such that 0 < c < 1/2, the probability of a random Cayley digraph on a group of order n and degree cn having diameter 2 is at least 2 We now turn to the case where k ≈ n δ with 1/2 < δ < 1.For k = λt with λ = o(1) as t → ∞, the approximation (11) is still valid, and we also have C(t, λ) = 1 + O(λ).Thus, if k grows at least as fast as n δ with δ > 1/2, for asymptotic computation we may replace k with 2n δ−1 t and set λ = 2n δ−1 .Then, the exponent tR(λ) in (10) may be replaced with −n 2δ−1 /2, which implies exponential decay if δ > 1/2.Using the lower bound of Proposition 3.3 again, we have the following conclusion.
Theorem 4.2 For every δ such that 1/2 < δ < 1, the probability of a random Cayley digraph on a group of order n and degree n δ having diameter 2 is at least 2 It is natural to ask what happens when k ≈ √ n.By the Moore bound for diameter two, the probability of a random Cayley digraph of degree k and order n (for a general group) having diameter 2 is zero if k < √ n .It is interesting to note that if the right-hand side is increased by a factor of 2, then for an arbitrary n of the form n = 2 2d there exists a Cayley (di)graph of order n and degree k = 2 √ n = 2 • 2 d on an elementary Abelian group of order n, which has diameter 2. Indeed, representing vertices of the graph as 2d-dimensional 0-1 vectors, it is sufficient to consider a generating set of the form S 1 ∪ S 2 where S 1 and S 2 consists of all non-zero vectors having the first d and the last d coordinates equal to zero, respectively.It follows that the probability that a random Cayley (di)graph on an elementary Abelian 2-group of order n and degree k = 2 √ n has diameter 2 is positive.This, of course, does not allow to make any conclusion as to how large this probability might be.
However, our approximation above shows that if k = c √ n , then the lower bound from Proposition 3.3 tends to negative infinity as n → ∞, while the upper bound for Abelian groups converges to 1 − exp(−c 2 /2).In particular, this shows that, when restricted to Cayley graphs on Abelian groups of order n and valence c √ n, the probability Pr(Diam ≤ 2) does not tend to 1 as n → ∞, for any value of c.This brings us to the question in the Introduction concerning the threshold for k = f (n) at which the asymptotic value of the upper bound on Pr(Diam ≤ 2) undergoes a phase transition, switching abruptly from 0 to 1 as k increases.Our previous findings allow us to give a more precise information about the transition.
Theorem 4.3 Let G be a finite group of order n and let P (n, k) denote the probability that a random Cayley digraph of degree k = f (n) on G has diameter at most 2.
Proof: To simplify the asymptotic calculations we may replace t with n/2 and λ = k/t with 2k/n.Moreover, by (11), (10) and other findings accumulated at the beginning of this Section, together with the fact that the coefficient at λ 3 in the O(λ 3 ) term in (11) is positive, for sufficiently large n and for k = o(n 2/3 ) we have an upper bound on a(n, k) of the form exp(−k 2 /(2n)) ≤ a(n, k) ≤ c 1 •exp(−k 2 /(2n)) for some positive constant c 1 that absorbs the multiplication effect of the term e tO(λ 3 ) = e O(k 3 /n 2 ) appearing in (11) and of the terms P (λ), C(t, λ) from (10).Combining this with Proposition 3.
where the positive constant c 2 < 1 absorbs the effect of multiplication by 1 − k/(n − 1) appearing in Proposition 3.3; note that for k = o(n 2/3 ) this constant can be chosen arbitrarily close to 1 for sufficiently large n.
Clearly it is sufficient to prove the statement in the case when µ has polynomial growth.Routine limit calculation using this assumption show that if k = 2n ln(c 1 nµ(n)) , then c 1 n exp(−k 2 /(2n)) • we will use the symbol G * y to denote the set G * \{y}.For each y ∈ G * let X (y) = {S ∈ G * y k : {x, x −1 y} ⊂ S for all x ∈ G * y }.We claim that M = χ • n − 2 k −1 where χ = max y∈G * |X (y)| .(6) Indeed, for each y ∈ G * , the distribution of S when the condition on the event [y / ∈ S] is uniform over the set G * y k .In particular, Pr(X(y) | y / ∈ S) = n−2 k −1 |X (y)|, and the claim now follows from the definition of M in (4).