THE ASYMPTOTIC NUMBER OF UNLABELLED
REGULAR GRAPHS
BELA BOLLOBAS
Over ten years ago Wright [4] proved a fundamental theorem in the theory of
random graphs. He showed that if M = M(n) is such that almost no labelled graph
of order n and size M has two isolated vertices or two vertices of degree n — 1, then
the number of labelled graphs of order n and size M divided by the number of
unlabelled graphs of order n and size M is asymptotic to n\. The result is best
possible, since if the above ratio is asymptotic to n\ then almost no labelled graph of
order n and size M has a non-trivial automorphism. The aim of this paper is to prove
the analogue of Wright's theorem for regular graphs.
Random regular graphs have not been studied for long. The main reason for this
is that until recently there was no asymptotic formula for the number of labelled
regular graphs: such a formula was found by Bender and Canfield [1]. Even more
recently, in [2] a model was given for the set of labelled regular graphs which makes
the study of random regular graphs fairly accessible. In particular, a result in [3]
implies that, for r ^ 3, almost every labelled r-regular graph has only the identity as
its automorphism.
Let r ^ 3 be fixed and let n -* oo in such a way that rn = 2m is even. Denote
by S£r = S£KtT the set of r-regular graphs with vertex set V = { l , 2 , . . . , n } . Write
°llr = °Unr for the set of unlabelled r-regular graphs of order n. Put Lr = \££?r\ and
Ur = \%\. We know from [1] that
^
Lr
(r2 _ 1)/4 (2m)!
m
K)
2 m\
•
The aim of this paper is to prove the following result.
THEOREM.
Ifr ^ 3 then
where m = rn/2.
Proof. The second relation is the result of Bender and Canfield [1] and so we
have to prove that Ur ~ LJn\. Throughout the proof all relations ( ~ , o, 0) refer to
the passage of n to oo and our inequalities are claimed to hold if n is sufficiently large.
We shall write c l 5 c 2 ,... for positive constants.
n
Let W = (J Wt be a fixed set with rn elements partitioned into n sets of r
elements each. Call a partition of W into m = rn/2 pairs a configuration. Clearly
there are N{m) = (2m)!/2mm! configurations. Given a configuration F, denote by
(f){F) the graph with vertex set V = {1, 2,..., n} in which ij is an edge if and only if F
Received 14 September, 1981.
J. London Math. Soc. (2), 26 (1982), 201-206
202
BELA BOLLOBAS
has a pair with one element in W{ and the other in Wy Clearly every G e 5£r is of the
form G = (j){F) for (r!)n configurations F, so that L, ^ N(m)/(r\)n. What we shall
need from the asymptotic formula for Lr is that if n is sufficiently large then
for some positive constant cx. (In fact every 0 < q < e~(r2~1)/4 will do.)
It is convenient to turn ifr into a probability space by giving all graphs G e S£r the
same probability, namely L~ 1 . Let us estimate the probability that a random graph
G e 5£r contains a fixed set of u edges. Clearly there are at most r2u sets of u pairs that
are mapped by c/> into our u edges. There are N(m — u) configurations that contain u
given pairs, and so the above probability is at most
= —r2uN{m-u)/N{m)
Pu = r^Nim-uyic^im)}
c
i
= —
r 2 "{(2m-l)(2m-3)...(2m-2M + l)}- 1
c
^ ir 2u ((2rn-2)!)- u/(2m - 1) ^ (^Y .
c
\
i
n
(2)
/
Let us now see what we have to show to prove (1). Consider the symmetric group
!„ acting on V. For co e Zn let i f (a>) be the set of graphs in ifr invariant under co and
put L((o) = \Se{w)\. Then
CU 6 I,,
C 6 if r
where a(G) is the order of the automorphism group of G. Since ifr has precisely
n\/a(G) graphs isomorphic to a given graph G e ifr,
Ur = £ ( M ) )
^
For the identity permutation 1 e I n we have L(l) = Lr so that (1) holds if and only if
X L(co)/L r = £ P(fi)) = o ( l ) ,
to el,,
co 6 I,,
(0^1
fO ^= 1
(3)
where P(co) = L(a>)/Lr is the probability that G e ifr is invariant under co.
We partition the sum in (3) into several parts. Denote by M(s,s2,s3) the set
of permutations in 2.n that move s vertices and have s2 2-cycles and s 3 3-cycles.
Note that if M{s,s2,s3) ± 0 then 2s2 + 2s3 ^ s. Furthermore, M(0, 0, 0) = {1},
M( 1,0,0) = 0 and, rather crudely,
\M(s,s2,s3)\^ns/{s2\s3\}.
(4)
The gist of our proof of (1) is an appropriate upper bound on
max {P{co): co e M(s, s2, s3)}. Though the proof of this upper bound is rather simple,
THE ASYMPTOTIC NUMBER OF UNLABELLED REGULAR GRAPHS
203
unfortunately it is somewhat cumbersome to present. We prepare the ground with a
simple but useful remark concerning representations of sets as unions of other sets.
Let s& be a collection of disjoint finite sets containing mi sets of size j ,
j = 1,2,..., t. Denote by f{b;stf) =f{b;m1,m2,...,mt)
the number of b-sets that
are unions of some of the sets in si. Then
(5)
where m'h = Y imjh
The equality is obvious, and so we prove the inequality. Let si be a collection of
disjoint sets consisting of nij >sets for h ^ j ^ t, but containing no j-sets for; < h.
For each At e s4 let C, be an /t-set so that ty = {C,: A{ e sJ) consists of disjoint
t
h-sets. Let %>" be a collection of mj, — £ m,- disjoint h-sets and b — hlb/h] singletons
h
so that ^ = ^" u # " consists of disjoint sets. We have to show that
f(b ; sJ) ^ /(fc ; ^ ) . To see this, suppose that a b-set B is the union of some sets in si'.
Let B' be a 6-set which is the union of those C, for which Ax a B and some sets in (€".
Then the map B -+ B' is one-to-one and so /(ft ; «s/) ^ f{b; ^ ) , proving (5).
Let us fix a permutation coe M(s,s2,s3), s ^ 2. We introduce the following
notation: S is the set of vertices moved by co, S 2 is the set of vertices in 2-cycles of
co, S 3 is the set of vertices in 3-cycles of co and, with a slight abuse of notation,
S 4 = S \ S 2 u S 3 . As before, we write s,- = |S£|/i so that s = 2s2 + 3s3 + 4s 4 . (Note
that s 4 need not be an integer.) We also put S1 = F \ S .
Let us consider the orbits of (the permutation induced by) co acting on those
elements of V(2), the set of unordered pairs of elements of V, that have at least one
element in S. For simplicity we say that an unordered pair (x, y) € V(2) joins Sk to St if
x e Sk and y eS,.
Denote by mh(k, I) the number of orbits of size h consisting of pairs joining Sk to
S,. By definition mh(k, 1) = mh(l, k). It is easily checked that
mh(k, 1) = 0 unless h ^ h(k, 1) = k ,
mh(3,4) = 0 unless h ^ h{3, 4) = 6 ,
w h (2,3) = 0 unless h = h{2, 3) = 6 ,
m h (2,2) = 0 unless /i = fi(2, 0) = 1,
m h (2,4) = 0 unless h ^ h{2,4) = 4 ,
or h = h{2,2) = 2 ,
mh(3,3) = O unless /i = h(3,3) = 3,
m h (4,4) = 0 unless A ^ li(4, 0) = 2 .
Futhermore,
m!(2,2)^s2
and
2m2(4, 4) + 3w 3 (4, 4) ^ 2s 4 .
To see the last statement note that if x is in a /c-cycle but xy belongs to an orbit of
size less than k, then k = 21 and y = colx, that is, the pair (x, y) joins diametrically
opposite vertices of an even cycle.
In order to give the notation some consistency we write h{3,0) = 1 and
h{4, 4) = 4. For 1 ^ k ^ /, / =£ 2, a {k, l)-orbit (or (/, k)-orbit) is an orbit of size at
least h{k, I) consisting of pairs joining Sk to St. Furthermore, for k ^ 2 a (k, 0)-orbit
204
BELA BOLLOBAS
is an orbit of size less than h{k,k) consisting of pairs joining Sk to Sk. The
(k, 0)-orbits are the exceptional orbits of pairs joining vertices of Sk in the sense that
altogether they contain at most ksJ2 = \Sk\/2 edges, while the (k; /c)-orbits have at
least ksk{ksk — 2)/2 edges.
Now we turn to our task of estimating P{co), a> e M{s1, s2, s3). If G e Le{(o) then
the set of edges of G cannot split orbits. Suppose that G has h(i,j)kij edges in the set
of (ij)-orbits. We call (fcy) the set of parameters of G. Note that these parameters
have the following properties:
k — k
h(i,j)kij is a non-negative integer ,
£ h(i,j)ktJ
+ h{i, 0)ki0 + h(i, i)ktl = rish
i = 2,3,4.
(6)
J= O
The last relation holds since an edge in an (i, j)-orbit contributes 1 to the sum of
degrees of the vertices in S{ unless) = 0 or i, when it contributes 2.
We shall call an array (k^) having the above properties a permissible array. When
(6) is written out in full, a permissible array (k^) satisfies
= 2rs2 ,
3/c31+6/c32 + 6/c33 + 6/c34 = 3rs 3 ,
4/c40 + 4/c41 + 4/c42 + 6/c43 + 8/c44 = 4rs 4 .
(6')
Given a permissible array (/c0) we write P(a>; (fc0)) for the probability that a graph
G e $£r belongs to J?(a>) and has (/c0) as its set of parameters. Then
P((D) = YP(w;(kij)),
(7)
where the summation is over all permissible arrays.
Let us estimate P(a>; (/c(J)) for a fixed permissible array (k^). Denote by s{i,j) the
number of pairs in the (i,))-orbits of a>. Clearly
s(i,j) ^ USiSj
with s0 defined to be 1. The set of h(i,j)kij edges consisting of full (i,j)-orbits can be
selected in at most
/(/,;)
=f(h(ij)kij;m1(ij),m2(i,j),...,mt(i,j))
ways. Here the inequality is a consequence of (5) and the fact that m,(i,j) = 0 for
/ < h(i,j). By (2) the probability of a graph G 6 ifr containing a given set of
(8)
THE ASYMPTOTIC NUMBER OF UNLABELLED REGULAR GRAPHS
205
edges incident with S is at most (cjnf. Hence
n),
(9)
where the product is over all unordered pairs {i,j) with max(ij) ^ 2.
We should like to bound from above the right-hand side of (9) as (k^) runs over
all permissible arrays. Though this is an elementary problem in optimization theory,
the manipulations in a standard solution are offputting so we prefer a different
approach.
We claim that
(c2/nfl\f(iJ)^n-s{i+s)$ss3
(10)
where e = 2" 4 . To prove (10) we shall separate the large elements k{J from the small
ones. Set
[1
iffcy^ KT 4 s/r,
£
u =
iffcy < 10" 4 s/r.
( 0
Then
k
{n/)K~ 1 0 ~ 2 s S2~ S 2 S3~ S 3 < ns-fcn££1^s££'^s2~S2s3~S3 = A,
say, where £ £\jkXi = ]T eljklj and the summation in the exponent of s is over
Recall that s - 2s2 + 3s3 + 4s4 so that by (6') and (8) we have
s ^ k = k20 ( - - l j + k2l ( - - 2 j + k22 ( - - 2
Hence if s = rf, by (6') we find that
log ,4
lo^"
logn ^
C3
s
ilogn
o^
4
+ s
"
+
£
£
iA;+a
J = 2
fi fc
2 ^I1 < j ^ y
4 y
- a X B2jh(2,j)k2j/(2r)-oi J e3jh{3J)k3J/{3r)
j=0
j=0
s
,
r-2
23
,
A
3 2
^
,2 4^( r ~ 2
))
£23all - - - - ) )
-*24(4—
2
a +- a
206
BELA BOLLOBAS
-1
\
,
/r-2
fc4
,
/ r-2
2a
k 4
0
logn
a\
[
r — 2 --, ,
r ^ J
s
logn
r—2
12
This implies (10) since 1/13- 10" 2 > 2" 4 .
Now the proof is easily completed. For a permutation co e M(s, s 2 , s3) there are
at most (6rs)12 permissible arrays and so by (7), (9) and (10)
P(CD)
Therefore by (4)
we M(s,52,s3)
w e I,,
w j= 1
s= 2
s= 2
This proves (3), and so the proof of our theorem is complete.
It is worth noting that the proof is considerably more pleasant for r ^ 5. In that
case we do not have to consider the number of 2-cycles and 3-cycles in a permutation
(D e !„; it suffices to use the number of vertices moved by co as the only parameter,
and to note that at most ns permutations move precisely s vertices.
In this paper we considered regular graphs but in fact the proof can be adapted
to graphs with more general degree sequences. In particular, the following result can
be obtained without any additional work. Let A ^ 3 be a constant. Then the number
of labelled graphs with degree sequences 3 ^ dY ^ d2 ^ ... ^ dn ^ A divided by the
number of unlabelled graphs having the same degree sequence is asymptotic t o n ! .
References
1. E. A. BENDER and E. R. CANFIELD, 'The asymptotic number of labeled graphs with given degree
sequences', J. Combin. Theory Ser. A, 24 (1978), 296-307.
2. B. BOLLOBAS, 'A probabilistic proof of an asymptotic formula for the number of labelled regular
graphs', European J. Combin., 1 (1980), 311-316.
3. B. BOLLOBAS, 'Distinguishing vertices of random graphs', Ann. Discrete Math., to appear.
4. E. M. WRIGHT, 'Graphs on unlabelled nodes with a given number of edges', Acta Math., 126 (1971),
1-9.
Department of Pure Mathematics and Mathematical Statistics,
University of Cambridge,
16 Mill Lane,
Cambridge CB2 1SB.