COMBINATRORICS AND
SHAPE CHARACTERISTICS
OF PLANAR (AND RELATED)
GRAPH CLASSES
Michael Drmota
Institute of Discrete Mathematics and Geometry
TU Wien
A 1040 Wien, Austria
[email protected]
http://www.dmg.tuwien.ac.at/drmota/
Symposium Diskrete Mathematik, Frankfurt, May 9–10, 2014.
Contents
• Planar Maps
• Planar Graphs
• Minor Closed Graph Classes
• Subcritical Graph Classes
Enumeration of Planar Maps
A planar map is a connected planar graph, possibly with loops and
multiple edges, together with an embedding in the plane.
A map is rooted if a vertex v and an edge e incident with v are distinguished, and are called the root-vertex and root-edge, respectively.
The face to the right of e is called the root-face and is usually taken
as the outer face.
Enumeration of Planar Maps
Mn ... number of rooted maps with n edges [Tutte]
Mn =
2(2n)!
3n
(n + 2)!n!
The proof is given with the help of generating functions and the socalled quadratic method.
Asymptotics:
Mn ∼ c · n−5/212n
Enumeration of Planar Maps
Root degree distribution
dn,k ... probability that the root-vertex has degree k in a map with n
edges ( = probability that the outer face has valency k in a map with
n edges – by duality)
dk = lim dn,k :
n→∞
dk ∼ c0 · k1/2
k
5
6
Enumeration of Planar Maps
Universal Property
Mn ∼ c · n−5/2γ n,
Class of maps
Arbitrary
Eulerian
3-connected
Loopless
2-connected
Bipartite
dk ∼ c0 · k1/2q k
γ
12
8
4
256/27
27/4
8
q
5/6
√
3/2
1/2
3/4
2/3
3/4
Enumeration of Planar Maps
Generating functions
Mn,k ... number of maps with n edges and outer-face-valency k
M (z, u) =
X
Mn,k uk z n
n,k
M (z, u) = 1 + zu2M (z, u)2 + uz
uM (z, u) − M (z, 1)
.
u−1
u ... “catalytic variable”
M =
M
M
+
M
Enumeration of Planar Maps
Completing the square leads to
h
i2
2
2
2u z(1 − u)M (z, u) − (1 − u + u z) = H(z, u, M (z, 1))
with
H(z, u, y) = 4(u − 1)u3z 2y + u4z 2 − 4u4z + 6u3z − 2u2z + u2 − 2u + 1.
General Form:
[G1(z, u, y(z))M (z, u) + G2(z, u, y(z))]2 = H(z, u, y(z))
where y(z) abbreviates M (z, 1).
Enumeration of Planar Maps
[G1(z, u, y(z))M (z, u) + G2(z, u, y(z))]2 = H(z, u, y(z))
Key observation:
∃ u(z) with H(z, u(z), y(z)) = 0
=⇒
Hu(z, u(z), y(z)) = 0
Quadratic Method
1. Solve the system H(z, u(z), y(z)) = 0,
Hu(z, u(z), y(z)) = 0
2. M (z, 1) = y(z)
3. M (z, u) =
q
H(z, u, y(z)) − G2(z, u, y(z)) /G1(z, u, y(z)).
Enumeration of Planar Maps
Planar Maps
ad 1. u = u(z) and y(z) = M (z, 1) are determined by
z=
(1 − u)(2u − 3)
,
2
u
M (z, 1) = −u
3u − 4
(2u − 3)2
ad 2. Elimination gives an equation for M = M (z, 1):
27z 2M 2 − 18zM + M + 16z − 1 = 0
and consequently
1
3/2
M (z, 1) = −
1
−
18z
−
(1
−
12z)
54z 2
=
2(2n)!
3n z n .
n≥0 (n + 2)!n!
X
ad 3.
q
M (z, u) =
H(z, u, M (z, 1)) + 1 − u + u2z
2u2z(1 − u)
.
Enumeration of Planar Maps
Planar Maps
Local expansions:
M (z, 1) = c0 + c2(1 − 12z) + c3(1 − 12z)3/2 + O((1 − 12z)2).
M (z, u) = c0(u) + c2(u)(1 − 12z) + c3(u)(1 − 12z)3/2 + O((1 − 12z)2),
=⇒
X
k≥1
2 −5/2 n
12
Mn ∼ √ n
π
√
u 3
[z n]M (z, u)
c3(u)
=
=q
n→∞ [z n ]M (z, 1)
c3
(2 + u)(6 − 5u)3
dk uk = lim
=⇒dk ∼ c · k1/2
k
5
6
Enumeration of Planar Maps
Theorem. (D. + Noy) Univeral property for catalytic equations
Let F (z, u) = n,k fn,k uk z n and y(z) = n ynz n with fn,k ≥ 0 and yn ≥ 0
be the unique solutions of the equation
P
P
[G1(z, u, y(z))F (z, u) + G2(z, u, y(z))]2 = H(z, u, y(z)),
(+ some technical conditions). Furthermore assume that
H(z0, y0, u0) = 0,
G1 6= 0,
Hy 6= 0,
Hu(z0, y0, u0) = 0,
Huy 6= 0,
Huuu 6= 0,
Huu(z0, y0, u0) = 0,
Hz Huy 6= Hy Hzu ,
2
Huuu Huy
− Hy Huy Huuuu + 3Hy Huuy Huuu 6= 0
Then there exists constants c0, c2, c3 with c3 6= 0 such that locally
y(z) = c0 + c2(1 − z/z0) + c3(1 − z/z0)3/2 + O((1 − z/z0)2).
Consequently we have (for some constant c > 0)
yn = [z n] y(z) ∼ c · n−5/2z0−n.
Enumeration of Planar Maps
Theorem (cont.)
Furthermore we have that for every integer k ≥ 0 the limit
dk = lim
fn,k
n→∞ yn
exists and we have, uniformly for k ≤ C log n,
fn,k
∼ c · q k k1/2
dn
for some c > 0, q = 1/u0 and any constant C > 0, in particular
dk ∼ c · k1/2q k .
2-Connected Planar Maps
B(z), B(z, u) ... GF of 2-connected rooted planar maps
M (z) = B(zM (z)2)
and
M (z, u) = B zM (z)2,
B
M
M =
M
M
M
M
M
M
M
M
M
uM (z, u)
M (z)
!
2-Connected Planar Maps
The resulting equations are slightly different but analytically they are
very similar to planar maps (algebraic of degree 2):
B(z, u)2 − (u2z + uB(z) + 1)B(z, u) + (u3 + u)B(z, 1)z + u2z − uz = 0.
[Tutte]
2(3n − 3)!
bn =
n!(2n − 1)!
Vertices of Given Degree
Theorem. (D.+Panagiotou 2013)
Xn ... number of vertices of degree k in connected or 2-connected
random planar maps with n edges
=⇒
with µ > 0 and σ 2 ≥ 0.
Xn − µn
→ N (0, σ 2)
√
n
Vertices of Given Degree
Extension of analytic treatment of catalytic equation with an
additional parameter w.
[G1(z, u, y(z, w), w)F (z, u, w) + G2(z, u, y(z, w), w)]2 = H(z, u, y(z), w),
=⇒
y(z, w) = c0(w) + c2(w)(1 − z/z0(w)) + c3(w)(1 − z/z0(w))3/2 + O((1 −
=⇒
yn(w) = [z n] y(z, w) ∼ c(w) · n−5/2z0(w)−n
=⇒
E wXn =
yn(w)
c(w)
z0(1)
∼
·
yn(1)
c(1)
z0(w)
!n
.
Consequently a Central Limit Theorem follows.
(The power (z0(1)/z0(w))n mimics a sum of independent random variables).
3-Connected Planar Maps
Bijection between connected maps and quadrangulations
3-Connected Planar Maps
Bijection between connected maps and quadrangulations
3-Connected Planar Maps
Bijection between connected maps and quadrangulations
3-Connected Planar Maps
Bijection between connected maps and quadrangulations
3-Connected Planar Maps
Bijection between connected maps and quadrangulations
3-Connected Planar Maps
In this bijection 2-connected maps correspond so quadrangulations,
where the outer face contains 4 vertices.
Moreover 3-connected maps correspond so simple quadrangulations of this type (every circle different from the outer circle of length
4 determines a face).
Quadrandgulations can be then recursively described by replacing every
face of a simple quadrangulation by a quadrangulation.
This leads (finally) to the generating function M3(x, y) for the number
of 3-connected edge-rooted planar maps according to the number
of vertices and edges
M3(x, y) = x2y 2
1
1
+
−1−
1 + xy
1+y
U = xy(1 + V )2,
(1 + U )2(1 + V )2
(1 + U + V )3
V = y(1 + U )2.
!
,
Quadrangulations
Schaeffer bijection: start with a quadrangulation
Quadrangulations
Schaeffer bijection: calulate the distance to the root vertex
0
1
2
2
1
2
1
1
2
4
3
2
Quadrangulations
Schaeffer bijection: there are only two possible constellations
k +2
k
k +1
k +1
k
k +1
k +1
k
Quadrangulations
Schaeffer bijection: include fat edges
k +2
k
k +1
k +1
k
k +1
k +1
k
Quadrangulations
Schaeffer bijection: include fat edges
0
1
2
2
1
2
1
1
2
4
3
2
Quadrangulations
Schaeffer bijection: include fat edges
0
1
2
2
1
2
1
1
2
4
3
2
Quadrangulations
Schaeffer bijection: delete the dotted edges
0
1
2
2
1
2
1
1
2
4
3
2
Quadrangulations
Schaeffer bijection: a labelled tree occurs
2
1
1
2
2
2
1
1
4
3
0
2
Quadrangulations
Schaeffer bijection: a labelled tree occurs
2
1
1
2
2
2
1
1
4
3
2
Well-Labelled Trees
Positive labels, root has label 1, adjacent labels differ at most by 1:
2
1
1
2
2
2
1
1
4
3
2
Well-Labelled Trees
Hn,k ... number of vertices of distance k from the root vertex in a
quadrangulation of size n
λn,k ... number of vertices with label k from in a well-labelled tree with
n edges
Theorem (Schaeffer)
There exists a bijection between edge-rooted quadrangulations with
n faces and well-labelled trees with n edges, such that the distance
profile (Hn,k )k≥1 of a quadrangulation is mapped onto the label distribution (λn,k )k≥1 of the corresponding well-labelled tree.
Continuous Limits
Theorem (Chassaing+Marckert)
Let rn denote the maximum distance from the root vertex in random
quadrangulations with n vertices:
=⇒
d
γn−1/4rn −→ RISE − LISE ,
where γ = 2−1/4.
Remark. The random variable RISE − LISE denotes the (random)
length of the support of the density if the ISE (= integrated superBrownian excursion).
Continuous Limits
A graph G with the graph distance dG is a metric space (G, dG)
Theorem (Le-Gall, Miermont)
Let Mn be a random planar map or a random quadrangulation. Then
Mn, c n−1/4dMn
d
−→ (M, dM) ,
where (M, dM) denotes the Brownian map (a random metric space)
and the convergence holds in the Gromov-Hausdorff topology.
(c is a positive constant.)
Corollary. Every parameter that depends continuously on the distance
has a limiting distribution (e.g. the diameter scaled by n−1/4).
Further Shape Characteristics of Planar Maps
(2)
Largest Block Mn
Theorem [Gao+Wormald, Banderier+Flajolet+Schaeffer+Soria]
(2)
Mn
(2)
− E Mn
n2/3
(2)
where E Mn
→ stable law (Airy distribution) ,
∼ (1/3) . . . n.
Maximum Degree
Theorem [Gao+Wormald]
P[|∆n − (log n + log log n)/ log 2| ≤ ωn] → 1 − O(1/ log n + 2−ωn ) ,
where ωn → ∞.
Planar Graph Enumeration
Planar Maps
connected → 2-connected ≡ quadrangulations → simple quadrangulations ≡ 3-connected
Whitney’s theorem
3-connected planar maps ≡ 3-connected planar graphs
Planar Graphs
3-connected → 2-connected → connected
T •(x, y) → B(x, y) → C(x, y)
Planar Graph Enumeration
Planar Graphs
∂B
∂C(x, y)
∂C(x, y)
= exp
,y
x
∂x
∂x
∂x
!!
,
∂B(x, y)
x2 1 + D(x, y)
=
,
∂y
2
1+y
T •(x, D)
x2 D
1+D
= log
1+y
T •(x, y) =
x2 y 2
2
!
xD2
−
,
1 + xD
1
1
+
−1−
1 + xy
1+y
U = xy(1 + V )2,
(1 + U )2(1 + V )2
(1 + U + V )3
V = y(1 + U )2.
!
,
Planar Graph Enumeration
3-connected planar graphs
M3(x, y) ... generating function for the number of 3-connected edgerooted planar maps according to the number of vertices and edges
Whitney’s theorem: every 3-connected planar graph has a unique
embedding into the plane.
=⇒ T •(x, y) = 12 M3(x, y): ... generating function for the number of
3-connected labelled edge-rooted planar graphs
Planar Graphs
Planar networks
A network N is a (multi-)graph with two distinguished vertices, called
its poles (usually labelled 0 and ∞) such that the (multi-)graph N̂
obtained from N by adding an edge between the poles of N is 2connected.
Let M be a network and X = (Ne, e ∈ E(M )) a system of networks
indexed by the edge-set E(M ) of M . Then N = M (X) is called the
superposition with core M and components Ne and is obtained by
replacing all edges e ∈ E(M ) by the corresponding network Ne (and,
of course, by identifying the poles of Ne with the end vertices of e
accordingly).
A network N is called an h-network if it can be represented by N =
M (X), where the core M has the property that the graph M̂ obtained
by adding an edge joining the poles is 3-connected and has at least 4
vertices. Similarly N = M (X) is called a p-network if M consists of 2
or more edges that connect the poles, and it is called an s-network if
M consists of 2 or more edges that connect the poles in series.
Planar Graph Enumeration
Planar networks
Trakhtenbrot’s canonical network decomposition theorem: any
network with at least 2 edges belongs to exactly one of the 3 classes
of h-, p- or s-networks. Furthermore, any h-network has a unique
decomposition of the form N = M (X), and a p-network (or any snetwork) can be uniquely decomposed into components which are not
themselves p-networks (or s-networks).
Planar Graph Enumeration
Planar networks
K(x, y) ... generating function corresponding to all planar networks
where the two poles are not connected by an edge.
D(x, y) ... generating function corresponding to all planar networks
with at least one edge
S(x, y) ... generating function corresponding to all s-networks
F (x, y) = D(x, y) − S(x, y) ... the generating function corresponding to
all non-s-networks (with at least one edge)
N (x, y) ... generating function corresponding to all non-p-networks.
B(x, y) ... generating function of 2-connected planar graphs.
Planar Graph Enumeration
Planar networks
x2
∂B(x, y)
=
K(x, y),
∂y
2
D(x, y) = (1 + y)K(x, y) − 1,
K(x, y) = eN (x,y),
S(x, y) = xD(x, y)(D(x, y) − S(x, y)),
T •(x, D(x, y))
= N (x, y) − S(x, y).
2
x D(x, y))
=⇒
T •(x, D)
1+D
xD2
= log
−
2
x D
1+y
1 + xD
∂B(x, y)
x2 1 + D(x, y)
=
∂y
2
1+y
!
Planar Graph Enumeration
The number of 2-connected planar graphs
Theorem [Bender, Gao, Wormald (2002)]
bn ... number of 2-connected labelled planar graphs
−7
2
bn ∼ c · n
γ2n n! ,
γ2 = 26.18...
Planar Graph Enumeration
Block-Decomposition
Planar Graph Enumeration
Block-Decomposition
Planar Graph Enumeration
Block-Decomposition
Planar Graph Enumeration
Generating Functions for Block-Decomposition
xC°
B°
xC°
B°
xC°
B°
xC°
xC°
xC°
xC°
•
•
C •(x) = eB (xC (x))
B •(x) = B 0(x), C •(x) = C 0(x),
Planar Graph Enumeration
Theorem [Gimenez+Noy (2005)]
gn .... number of all labelled planar graphs
−7
2
gn ∼ c · n
γ n n! ,
γ = 27.22...
Random Planar Graphs
Degree Distribution
Theorem [D.+Giménez+Noy, Panagiotou+Steger]
Let dn,k be the probability that a random node in a random planar
graph Rn has degree k. Then the limit
dk := lim dn,k
n→∞
exists. The probability generating function
p(w) =
X
dk w k
k≥1
−1
can be explicitly computed; dk ∼ c k 2 q k for some c > 0 and 0 < q < 1.
d1
0.0367284
d2
0.1625794
d3
0.2354360
d4
0.1867737
d5
0.1295023
d6
0.0861805
Random Planar Graphs
• Implicit equation for D0(y, w):
√
S(D0 (t − 1) + t)
−
4(3t + 1)(D0 + 1)
D02 (t4 − 12t2 + 20t − 9) + D0 (2t4 + 6t3 − 6t2 + 10t − 12) + t4 + 6t3 + 9t2
,
−
4(t + 3)(D0 + 1)(3t + 1)
1 + 2t
1 t2 (1 − t)(18 + 36t + 5t2 )
where t = t(y) satisfies y+1 =
.
exp −
(1 + 3t)(1 − t)
2 (3 + t)(1 + 2t)(1 + 3t)2
and S = (D0 (t − 1) + t)(D0 (t − 1)3 + t(t + 3)2 ).
1 + D0 = (1 + y w ) exp
• Explicit expressions in terms of D0(y, w) (SEVERAL PAGES !!!!):
B0(y, w), B2(y, w), B3(y, w)
• Explict expression for p(w):
p(w) = −eB0(1,w)−B0(1,1)B2(1, w) + eB0(1,w)−B0(1,1)
1 + B2(1, 1)
B3(1, w)
B3(1, 1)
Maximum Degree
Maximum degree ∆n in random planar graphs
Theorem [D.+Gimenéz+Noy+Panagiotou+Steger]
P[∆n = c log n + O(log log n)] → 1
for c ≈ 2.529464248....
Remark. [McDiarmid+Reed (2008)]: c1 log n ≤ E ∆n ≤ c2 log n w.h.p.
Largest Block
(2)
Maximum block size Mn
in random planar graphs
Theorem [Panagiotou+Steger, Noy+Rue]
(2)
Mn
(2)
− E Mn
n2/3
(2)
where E Mn
→ stable law (Airy distribution) ,
∼ 0.959 . . . n.
Diameter
Diameter Dn in random planar graphs
Theorem [Chapuy+Fusy+Gimenéz+Noy]
P[Dn ∈ (n1/4−ε, n1/4+ε)] → 1
for every ε > 0.
Planar Graphs
Open Problems
• Continuous limit (to Brownian map) ???
• Asymptotic enumeration of unlabelled planar graphs ???
• CLT for the number of vertices of given degree ???
Maximum Degree for Random Planar Graphs
P[∆n = c log n + O(log log n)] → 1
Proof Strategy
1. Establish generating functions for dn,k
2. Analytic structure of generating functions
3. Upper bound with First Moment Method
4. Lower bound with Boltzmann Sampling
Maximum Degree for Random Planar Graphs
Generating functions for the degree distribution of planar graphs
C • = ∂C
∂x ... GF, where one vertex is marked
x ... vertices
y ... edges
w ... additional variable that counts the degree of the marked vertex
Generating functions:
C •(x, y, w)
connected rooted planar graphs
B •(x, y, w)
2-connected rooted planar graphs
T •(x, y, w)
3-connected rooted planar graphs
Maximum Degree for Random Planar Graphs
w
•
•
•
C (x, y, w) = exp B xC (x, y, 1), y, w ,
1
∂B •(x, y, w)
T•
D(x, y, w)
x, D(x, y, 1),
D(x, y, 1)
= xyw exp S(x, y, w) + 2
x D(x, y, w)
1
×
D(x, y, w) = (1 + yw) exp S(x, y, w) + 2
x D(x, y, w)
!!
D(x, y, w)
× T • x, D(x, y, 1),
−1
D(x, y, 1)
S(x, y, w) = xD(x, y, 1) (D(x, y, w) − S(x, y, w)) ,
∂w
2y 2w2
x
T •(x, y, w) =
2
1
1
+
− 1−
1 + wy
1 + xy
−
q

(u + 1)2 −w1(u, v, w) + (u − w + 1) w2(u, v, w) 
2w(vw + u2 + 2u + 1)(1 + u + v)3
u(x, y) = xy(1 + v(x, y))2,
v(x, y) = y(1 + u(x, y))2.
,

!!
Maximum Degree for Random Planar Graphs
with polynomials w1 = w1(u, v, w) and w2 = w2(u, v, w) given by
w1 = − uvw2 + w(1 + 4v + 3uv 2 + 5v 2 + u2 + 2u + 2v 3 + 3u2 v + 7uv)
+ (u + 1)2 (u + 2v + 1 + v 2 ),
w2 =u2 v 2 w2 − 2wuv(2u2 v + 6uv + 2v 3 + 3uv 2 + 5v 2 + u2 + 2u + 4v + 1)
+ (u + 1)2 (u + 2v + 1 + v 2 )2 .
Maximum Degree for Random Planar Graphs
... cancellation of the √-term in the singular expansion leads to
B 0(x, y) = g1(x, y) + h1(x, y)
x
1−
R(y)
!3
2
and through the critical equation
0
0
C 0(x, y) = eB (xC (x,y),y)
we obtain
C 0(x, y) = g2(x, y) + h2(x, y)
=⇒
x
1−
r(y)
!3
2
x
C(x, y) = g3(x, y) + h3(x, y) 1 −
r(y)
=⇒
7
−2
−n
cn ∼ c r(1) n n!
!5
2
Maximum Degree for Random Planar Graphs
Similarly, if w is taken into account
B •(x, y, w) = g1(x, y, w) + h1(x, y, w)
x
1−
R(y)
!3
2
,
where the functions w 7→ g1(R(1), 1, w) and w 7→ h1(R(1), 1, w) analytic
for w < w0 and have an algebraic singularity at w = w0.
With
•
•
0
C (x, 1, w) = exp B xC (x), 1, w
we obtain
=⇒
C •(x, y, w) = g2(x, y, w) + h2(x, y, w)
x
1−
r(y)
!3
2
Maximum Degree for Random Planar Graphs
connected planar graphs
[xnwk ]C •(x, 1, w)
dn,k =
[xn]C •(x, 1, 1)
[xn]C •(x, 1, 1) ∼ c1n−5/2ρ−n
C .
[xnwk ]C •(x, 1, w) ≤ w0−k [xn]C •(x, 1, w0)
∼ w0−k c2n−5/2ρ−n
C
=⇒ dn,k = O(w0−k ) = O(q k )
for q = 1/w0.
Maximum Degree for Random Planar Graphs
First moments
dn,k ... probability that a random vertex in Gn has degree k
(k)
Xn
.... number of of vertices in Gn of degree k
(k)
E Xn = n dn,k
∆n > k ⇐⇒
X
(`)
Xn
>0
`>k

=⇒



X (`)
X (`)
X




P[∆n > k] = P
Xn > 0 ≤ E
Xn
=n
dn,` = O(nq k )
`>k
`>k
`>k
=⇒ P{∆n ≤ c log n + r} ≤ 1 − O(q r )
Minor Closed Graph Classes
A graph class G is minor closed if every minor of a graph in G belongs
to G, too.
Theorem. [Robertson, Seymour]
Every minor closed graph class G of graphs is defined in terms of a
finite number of excluded minors.
Notation. Ex(H1, . . . , Hk ).
Examples. Planar graphs: Ex(K5, K3,3), Series-parallel graphs: Ex(K4)
Graphs that are embeddable into a surface of given genus g.
Graphs of tree-width ≤ t.
General Property Gn ≤ cnn! [Norine+Seymour+Thomas+Wollan]
Gn denotes the number graphs of size n in a vertex labelled minor
closed class.
Minor Closed Graph Classes
A graph class G is addable if all connected components of a graph in G
are in G, too, and if we can connect two vertices of different connected
components of a graph in G and the resulting graph is in G, too.
Theorem (McDiarmid)
For an addable minor closed graph class (of vertex labelled graphs) the
limit
Gn/n!
=γ
lim
n→∞ G
n−1 /(n − 1)!
exists and is finite.
Minor Closed Graph Classes
We say that a graph property can be formulated in monaic second
order logic (MSO) if we can quantify over sets of vertices.
For example, connectivity or k-colorability can be expressed in MSO.
Theorem (Heinig+Müller+Noy+Taraz)
Let G be an addable minor closed graph class. Then for every MSO
formula φ there exists a liminting probability p(φ). Moreover, if we
restrict to connected graphs in G then p(φ) ∈ {0, 1}.
Minor Closed Graph Classes
Graphs that are embeddable into a genus g (orientable) manifold
Theorem [Chapuy+Fusy+Gimenez+Mohar+Noy, Bender+Gao]
(g)
The numbers cn of vertex labelled connected graphs of size n that can
be embedded into an orientable manifold of genus g is asymptotically
given by
(g)
cn
∼ Cg n5(g−1)/2−1γ nn! ,
where Cg > 0.
Remark. γ = 27.2269 does not depend on g.
Remark. There is a similar result for non-orientable surfaces.
Subcritical Graph Classes
block: 2-connected component
Block-stable conneceted graph class G: G contains the one-edge
graph and all 2-connected components of a (connected) graph G ∈ G.
Equivalently, the 2-connected graphs of G and the one-edge graph
generate freely all graphs of G.
Examples: Planar graphs, series-parallel graphs, minor-closed graph
classes etc.
B(x) ... GF for 2-connected graphs in G
C(x) ... GF for connected graphs in G
Subcritical Graph Classes
xC°
B°
xC°
B°
xC°
B°
xC°
xC°
xC°
xC°
•
•
C •(x) = eB (xC (x))
B •(x) = B 0(x) ... GF for vertex-pointed 2-connected graphs
C •(x) = C 0(x) ... GF for vertex-pointed connected graphs
Subcritical Graph Classes
Definition.
C •(x).
Let ρB and ρC the radii of convergence of B •(x) and
The corresponding (block-stable) class G is called subcritial if
ρC C •(ρC ) < ρB .
Remark. x = ρC C •(ρC ) satisfies xB 00(x) = 1. Hence beeing subcritial
is equivalent to
ρB B 00(ρB ) > 1.
Subcritical Graph Classes
Lemma. Suppose that B(x) has radius of convergence ρB ∈ (0, ∞].
lim B 00(x) = ∞
x→ρB
=⇒
subcritical.
Corollary If B •(x) is entire or has a squareroot singularity:
s
x
B •(x) = g(x) − h(x) 1 − ,
η
then we are in the subcritical case.
Subcritical Graph Classes
• Trees are subcritical
• Cacti graphs are subcritical
• Outerplanar graphs are subcritical
• Series-parallel graphs are subcritical
• Planar graphs are critical
Labeled Trees
Rooted Trees:
1
B •(x) = x
R(x) = xC •(x) ... generating function of rooted, labelled trees
• (xC • (x))
•
B
C (x) = e
=⇒ R(x) = xeR(x)
Remark: T (x) ... GF for unrooted labelled trees:
1
0
T (x) = R(x)
x
=⇒
1
T (x) = R(x) − R(x)2
2
Cacti Graphs
21
25
34
32
26
8
4
12
16
17
10
20
1
28
22
14
27
11
33
3
15
31
29
6
2
19
13
9
5
24
18
7
23
30
Cacti Graphs
Generating functions
•
•
C •(x) = eB (xC (x)) ,
2
3
2
x
x
2x
−
x
B •(x) = x +
+
+ ··· =
.
2
2
2(1 − x)
2-connected cactus graph = edge or cycle
Outerplanar Graphs
21
25
34
32
26
8
4
12
16
17
10
20
1
28
22
14
27
11
33
3
15
31
29
6
2
19
13
9
5
24
18
All vertices are on the infinite face.
7
23
30
Outerplanar Graphs
Generating functions
•
•
C •(x) = eB (xC (x)) ,
1 + 5x −
B •(x) =
q
1 − 6x + x2
.
8
2-connected outerplanar graphs = dissections of the n-gon
Series-Parallel Graphs
Series-parallel extension of a tree or forest
Series-extension:
Parallel-extension:
Series-Parallel Graphs
Generating functions
bn,m ... number of 2-connected labelled series-parallel graphs with
n vertices and m edges
xn m
y
B(x, y) =
bn,m
n!
n,m
X
cn,m ... number of connected labelled series-parallel graphs with n
vertices and m edges
xn m
C(x, y) =
cn,m
y
n!
n,m
X
Series-Parallel Graphs
Generating functions
∂C(x, y)
∂B
∂C(x, y)
= exp
,y
x
∂x
∂x
∂x
!!
x2 1 + D(x, y)
∂B(x, y)
=
,
∂y
2
1+y
D(x, y) = (1 + y)eS(x,y) − 1,
S(x, y) = (D(x, y) − S(x, y))xD(x, y).
,
Critical vs. Subcritical Graphs
What does “subcritical” mean?
In a subcritical graph class the average size of the 2-connected
components is bounded.
=⇒ This leads to a tree like structure.
=⇒ The law of large numbers should apply so that we can expect
universal behaviours that are independent of the the precise structure
of 2-connected components.
Critical graph classes are notoriously more difficult to analyze and
we cannot expect universal laws.
Unlabelled Graph Classes
Cycle index sums
ZG (s1, s2, . . .) :=
X 1
X
n n! σ,g∈Sn ×Gn
σ·g=g
c (σ) c2 (σ)
c (σ)
s2
· · · snn
s11
where cj (σ) denotes the number of cycles of size j in σ ∈ Sn
G(x) = ZG (x, x2, x3, · · · )
∂
ZG (s1, s2, . . .)
ZG • (s1, s2, . . .) =
∂s1
G•(x) = ZG • (x, x2, x3, · · · ) =
∂
ZG (x, x2, x3, · · · )
∂s1
Unlabelled Graph Classes
Block decomposition

C •(x) = exp 
X 1
i≥1 i

ZB • (xiC •(xi), x2iC •(x2i), . . .)
• Dichotomy between subcritical and critical can be defined in a
natural way.
• Unlabelled trees are subcritical.
• Unlabelled cacti graphs are subcritical
• Unlabelled outerplanar graphs are subcritical
• Unlabelled series-parallel graphs are subcritical.
Subcritical Graph Classes
Universal properties
• Asymptotic enumeration:
Labeled case:
cn ∼ c0 n−5/2ρ−nn!
Unlabelled case:
cn ∼ c00 n−5/2ρ−n
(c0, c00 > 0, ρ ... radius of convergence of C(z))
[D.+Fusy+Kang+Kraus+Rue 2011]
Additive Parameters in Subcritical Graph Classes
Theorem [D.+Fusy+Kang+Kraus+Rue]
Xn ... number of edges / number of blocks / number of cut-vertices
/ number of vertices of degree k
=⇒
Xn − µn
→ N (0, σ 2)
√
n
with µ > 0 and σ 2 ≥ 0.
Remark. There is an easy to check “combinatorial condition” that
ensures σ 2 > 0.
Extremal Parameters in Subcrit. Graph Classes
(2)
• Maximum block size Mn
Theorem [D.+Noy, Panagiotou+Steger]
(2)
E Mn
= O(log n)
(2)
If the limit lim bn+1/(nbn) exists and is positive then E Mn is of order
log n and the deviation from the mean is a disrete version of the Gumbel
distribution:
P[Mn(2) ≤ k] ∼ exp (− exp(log n − f (k)))
uniformly for n, k → ∞, where f (k) is a function with f (k) ∼ ck for
some constant c > 0.
Extremal Parameters in Subcrit. Graph Classes
• Diameter Dn
Theorem [D.+Noy]
√
p
c1 n ≤ E Dn ≤ c2 n log n
• Maximum degree ∆n
Theorem [D.+Noy]
log n
c1
≤ E ∆n ≤ c2 log n
log log n
Continuum Limit
Theorem (Panagiotou+Weller)
Let Gn be a random graph in a subcritical graph family. Then
Gn, cn−1/2dGn
d
−→ (T , dT ) ,
where (T , dT ) denotes the Continuum random tree and the convergence holds in the Gromov-Hausdorff topology.
(c is a positive constant.)
√
Corollary. The scaled diameter Dn/ n has a univeral limiting distribution.
Subcritical Graph Classes
Open Problem
• A minor closed graph class Ex(H1, . . . , Hk ) is sub-critical if and only
if one of the Hj is planar ????
Thank You!