Random graphs with
forbidden vertex degrees
Abstract.
We study the random graph conditioned on the event that all vertex degrees lie in some given subset of the nonnegative integers. Subject to a certain hypothesis on , the empirical distribution of the vertex degrees is asymptotically Poisson with some parameter given as the root of a certain ‘characteristic equation’ of that maximises a certain function . Subject to a hypothesis on , we obtain a partial description of the structure of such a random graph, including a condition for the existence (or not) of a giant component. The requisite hypothesis is in many cases benign, and applications are presented to a number of choices for the set including the sets of (respectively) even and odd numbers. The random even graph is related to the randomcluster model on the complete graph .
Key words and phrases:
Random graph, even graph, randomcluster model.2000 Mathematics Subject Classification:
05C80, 05C071. Introduction
Let be a fixed nonempty set of nonnegative integers. The purpose of this paper is to study the structure of random graphs having all their vertex degrees restricted to the set .
We call a graph an graph if all its vertex degrees belong to . For example, if is a singleton, an graph is the same as a regular graph of degree . (We are not going to say anything new about this case.) One of our main examples is the class of Eulerian graphs, or even graphs, given by the set of even numbers , with the set of nonnegative integers. See Section 6 for further examples.
More precisely, we will study the random graph defined as conditioned on being an graph, where is the standard random subgraph of the (labelled) complete graph where two vertices are joined by an edge with probability , and these events, corresponding to the edges of , are independent. In other words, is a random subgraph of such that, if is any given subgraph of that is an graph, then
(1.1) 
where is the number of edges of . We are interested in asymptotics as , and we will tacitly consider only such that there exists an graph with vertices, in other words, such that the denominator in (1.1) is nonzero. Thus, a finite number of small may be excluded; moreover, if contains only odd integers, then has to be even. (It is easy to see that, apart from this parity restriction, all large are allowed.)
Remark 1.1.
The choice gives a random graph that is uniformly distributed over all graphs on labelled vertices. However, we will in this paper instead study the case when is of order and the average vertex degree is bounded (for , and as we shall see later, for too).
It follows immediately from (1.1) that two subgraphs of with the same degree sequence are attained with the same probability. Hence, the conditional distribution of given the degree sequence is uniform. We will therefore focus on studying the random degree sequence of ; it is then possible to obtain further results on the structure of by applying standard results on random graphs with given degree sequences to conditioned on the degree sequence. For example, using the results by Molloy and Reed [14, 15] we obtain Theorem 3.1 below on existence of a giant component in .
2. Main theorem
By symmetry, the labelling of the vertices and thus the order of the degree sequence is not important, and we shall therefore study the numbers of vertices with given degrees, rather than the degree sequence itself. We introduce some notation.
Let be the set of sequences of nonnegative integers with only a finite number of nonzero terms . Let
the set of such sequences supported on . For a (multi)graph , let be the number of vertices of degree in , , and let be the sequence of degree counts. Thus, is an graph if and only if . Clearly, cf. (1.1),
(2.1) 
We shall call this summation the partition function, denoted as .
Let be the set of probability distributions on . In other words, is the set of sequences of nonnegative real numbers such that . We regard as a topological space with the usual topology of weak convergence (denoted ); it is well known that this topology on may be metrised by the total variation distance
If has vertices, let , the proportion of vertices of degree , and . Note that is the probability distribution of the degree of a randomly chosen vertex in .
Let be the exponential generating function of ,
(2.2) 
Note that this is an entire function of , and that for while if and only if .
Let be the distribution of a distributed variable given that it belongs to , i.e., with . Thus, recalling (2.2),
(2.3) 
This conditional distribution is always defined for , and in the case for too (in which case it is a point mass at 0). The mean of the distribution is
(2.4) 
Let . We shall refer to the equation
(2.5) 
as the characteristic equation of the set (for this value of ), and we write
(2.6) 
where we allow only if . We further define the auxiliary function
(2.7) 
and note that when , equals the simpler function
(2.8) 
All logarithms in this paper are natural. We let denote positive constants, generally depending on and (or ) and sometimes on other parameters too (but not on ), which may be indicated by arguments. We sometimes assume that to avoid trivialities.
Theorem 2.1.
Let and suppose that contains a unique that maximizes (or, equivalently, ) over . Then, the following hold, as :

. In other words, for every ,
(2.9) 
All moments of the random distribution converge to the corresponding moments of . In other words, if is the degree sequence of , and , then for every ,
(2.10) In particular,
(2.11) 
The error probabilities in (i) decay exponentially: for every , there exists a constant such that, for all large ,
(2.12) 
We have that
(2.13)
More generally, let be the subset of where (or ) is maximal:
(2.14) 
If contains a single element, we thus take that element as ; in particular, if , then . We shall see in Section 4 that this is the normal case: is always finite and nonempty, and except for at most a countable number of values of .
Further results on and the auxiliary functions are given in Section 4.
Remark 2.2.
The set may contain more than one element, see Example 6.8. In this case, the theorem may not be applied, but the proof in Sections 8–9 extends to show that the random degree distribution approaches the finite set in the sense that the analogue of (2.12) holds for the distance to this set, i.e.,
(2.15) 
We can regard the distributions in as pure phases, in analogy with the situation for many infinite systems of interest in statistical physics, but for the finite systems considered here this has to be interpreted asymptotically. Thus, for large , the degree distribution of is approximately given by one of the pure phases, but we do not know which one. It follows that if we let , one of the following happens for the random degree distribution (regarded as an element of ):

converges in probability to for some .

converges in distribution to some nondegenerate distribution on . (A mixture of two or more pure phases.)

There are oscillations and does not converge in distribution; suitable subsequences converge as in (i) or (ii), but different subsequences may have different limits.
It is easy to show by a continuity argument that all three cases may occur in Example 6.8 for suitable sequences (with defined there). We do not know whether all three cases may occur for fixed .
We shall not investigate the case further here.
Remark 2.3.
We close this section with an informal explanation of the results of Theorem 2.1, several applications of which are presented in Section 6. Recall the partition function of (2.1), considered as a summation over suitable graphs. We wish to establish which graphs are dominant in this summation. In so doing, we will treat certain discrete variables as continuous, and shall study maxima by differentiation and Lagrange multipliers. Let be a sequence of nonnegative reals satisfying , and for . We write
Let represent the contribution to the summation of (2.1) from graphs having, for each , approximately vertices with degree . The (empirical) mean vertexdegree of such a graph is .
Now, is a summation over simple graphs subject to constraints on the vertex degrees. It may be approximated by a similar summation over certain multigraphs, and this is easier to express in closed form, as follows. The number of ways of partitioning vertices into sets of respective sizes , , is
Each vertex will be taken to have degree , and we therefore provide with ‘halfedges’. Each such halfedge will be connected to some other halfedge to make a whole edge. Since halfedges are considered indistinguishable, we shall require the multiplicative factor
The total number of halfedges is , and we assume for simplicity that is an integer. These halfedges may be paired together in any of
ways, and each such pairing contributes
to . We combine the above to obtain an approximation to :
By Stirling’s formula, as ,
(2.16) 
We maximize the last expression subject to to find that
(2.17) 
for some constant and some satisfying
(2.18) 
Thus is the mass function of the distribution and, by (2.4) and the definition of ,
(2.19) 
We combine this with (2.18) to obtain the ‘characteristic equation’ (2.5).
If there exists a unique satisfying the characteristic equation, then we are done. If there is more than one, we pick the value that maximizes the right hand side of (2.16). That is to say, the exponential asymptotics of are dominated by the contributions from graphs with satisfying (2.17) with chosen to satisfy the characteristic equation and to maximize .
3. The giant and the core
We show next how to apply Theorem 2.1, in conjunction with results of Molloy and Reed [14, 15] and Janson and Luczak [10, 11], to identify the sizes of the giant cluster and the core of . The proofs are deferred to Section 10.
We consider first the existence or not of a giant component in the random graph as . Let be a vector of nonnegative reals with sum 1, and write . As explained in [14; 15], if we consider the random graph with given degree sequence , and assume that there are vertices with degree , the quantity that is key to the existence of a giant component is
Subject to certain conditions, if , there exists a giant component, while there is no giant component when .
We shall apply this with , and to that end we introduce some further notation. Let, see (2.4),
(3.1)  
(3.2)  
(3.3) 
Note that . Furthermore, the only possibly negative term in the sums in (3.2) and (3.3) for are those with , while the terms with and always vanish and the others are positive unless .
Let be the component of with the largest number of vertices, and let be the second largest. (Break ties by any rule.) We write for the number of vertices in a graph .
Theorem 3.1.
Suppose that . Let and suppose that contains a unique element . Then, has a giant component if and only if , i.e., if and only if . More precisely, as ,
where
(3.4)  
(3.5) 
with given as follows:

if and , then is the unique solution to with , and ;

if and , then and , ;

if , then and .
Remark 3.2.
If , then and we are in Case (iii) with no giant component; in fact, by Theorem 2.1, so almost all vertices are isolated. In this case for all .
If , then on and on in all three cases, as follows from [11, Lemma 5.5], which yields another characterization of .
Remark 3.3.
If , then as soon as . Hence we are in Case (ii) if and in Case (iii) in .
Remark 3.4.
Remark 3.5.
It is easily seen, using (3.3), that equals the extinction probability of a Galton–Watson process with offspring distribution
that is, the distribution where . (Note that .) Hence , the asymptotic relative size of , equals by (3.4) the survival probability of a Galton–Watson process with offspring distribution and initial distribution .
The core of a graph is the largest induced subgraph having mimimum vertex degree at least . The core of an Erdős–Reńyi random graph has attracted much attention; see [10] and the references therein. Theorem 2.1 may be applied in conjunction with Theorem 2.4 of Janson and Luczak [10] to obtain the asymptotics of the core of . Let denote the core of . We shall require some further notation in order to state our results for .
Let . Let , and let be a random variable with the distribution. For , let be obtained by ‘thinning’ at rate so that, conditional on , has the binomial distribution . For , let
Theorem 3.6.
Let and suppose that contains a unique element . Let , and let, with as above,
As :

if ,
if, further, , then

if , and in addition on some nonempty interval , then
Remark 3.7.
Let be the Galton–Watson process with offspring distribution , started with a single individual , and let be the modified process where the first generation has distribution , cf. Remark 3.5. It may be seen that is the probability that the family tree of contains an infinite subtree with root and every node having children. Similarly, equals the probability that contains an infinite regular subtree with root (the root has children and all other vertices have ). It is easy to see heuristically that this yields the asymptotic probability that a random vertex belongs to the core, see Pittel, Spencer and Wormald [16] (for ), but it is difficult to make a proof based on branching process theory; see Riordan [17] where this is done rigorously for another random graph model.
4. Roots of the characteristic equation
To avoid some trivial complications, we assume throughout this section that , thus excluding the trivial case for which comprises isolated vertices only.
Lemma 4.1.
for all .
Proof.
Theorem 4.2.
For each , the set is finite and nonempty.
Proof.
The characteristic equation (2.5) may be written as where . By Lemma 4.1, for ,
(4.1) 
and thus for .
Since is an entire function, and does not vanish identically by what we just have shown, it has only finitely many zeros in each bounded subset of the complex plane, and in particular in the interval . Hence is finite.
To see that is nonempty, let be the smallest element of . If , then . If , then as , so for small positive . Since further is negative for large , possesses a zero on the positive real axis. ∎
We have defined as the maximum point of or on . The next theorem shows that, alternatively, it can be defined as the maximum point of on (but not of ). Furthermore, instead of , we can use the function
(4.2) 
that arises as follows. In Section 7, we will indicate the use of multigraphs in proving Theorem 2.1, of which we shall derive a multigraph equivalent at Theorem 7.3. With denoting the multigraph partition function, we shall see in the proof of Theorem 7.3 that represents the contribution to from multigraphs with degree distribution close to , see Remark 8.3. For this reason, is a more natural function than , although it has a more complicated formula. We shall have to exclude the trivial case when is a singleton; in this case is constant.
It is easily seen that for all and , with equality if and only if .
We regard and as functions of , with considered a fixed parameter. These functions are evidently analytic on . Note that if , then and , and are continuous at with . On the other hand, if , then and while a simple calculation yields , where . as
Theorem 4.3.
The following hold for every fixed and or , except for in the trivial case .
(i) is the set of stationary points of , possibly with added:
(ii) is the set of global maximum points of :
Proof.
(i): Differentiation yields
(4.3) 
and, after some simplifications,
(4.4) 
By the Cauchy–Schwarz inequality, provided and ,
(4.5) 
(See Theorem 5.2 below for a more general result.) By (4.3)–(4.5), for , if and only if , i.e., the characteristic equation (2.5) holds.
(ii): By (4.3)–(4.5) and Lemma 4.1, is decreasing for large . Furthermore, by the remarks prior to the theorem, is either continuous at 0 or tends to there. This implies that has a finite maximum, attained at one or several points in . It remains to show that the maximum points belong to ; it then follows that equals the set of maximum points.
If is a maximum point of , then and by (i).
We define and ; thus (and Theorem 2.1 applies) if and only if , and in that case . We have defined only when ; for convenience we extend the definition to all by letting by any element of (for example or ).
Corollary 4.4.
For every and ,
Theorem 4.5.
If , then , with equality only if .
Proof.
If , then