[PPT] - Enumeration and uniform sampling of planar structures Mihyun Kang PowerPoint Presentation

SLIDE 1

Enumeration and uniform sampling

f planar structures

Mihyun Kang Institut für Informatik Humboldt-Universität zu Berlin

SLIDE 2

Planar structures

Planar structures are classes of graphs that are embeddable in the plane:

Trees : K3 minor-free graphs
Planar graphs : K5, K3,3 minor-free graphs (Kuratowski’s theorem)
Series-parallel graphs : K4 minor-free graphs
Outerplanar graphs : K4, K2,3 minor-free graphs
· · ·

SLIDE 3

Planar structures

Planar structures are classes of graphs that are embeddable in the plane:

Trees : K3 minor-free graphs
Planar graphs : K5, K3,3 minor-free graphs (Kuratowski’s theorem)
Series-parallel graphs : K4 minor-free graphs
Outerplanar graphs : K4, K2,3 minor-free graphs
· · ·

SLIDE 4

Planar structures

Planar structures are classes of graphs that are embeddable in the plane:

Trees : K3 minor-free graphs
Planar graphs : K5, K3,3 minor-free graphs (Kuratowski’s theorem)
Series-parallel graphs : K4 minor-free graphs
Outerplanar graphs : K4, K2,3 minor-free graphs
· · ·

SLIDE 5

Planar structures

Planar structures are classes of graphs that are embeddable in the plane:

Trees : K3 minor-free graphs
Planar graphs : K5, K3,3 minor-free graphs (Kuratowski’s theorem)
Series-parallel graphs : K4 minor-free graphs
Outerplanar graphs : K4, K2,3 minor-free graphs
· · ·

SLIDE 6

Planar structures

Planar structures are classes of graphs that are embeddable in the plane:

Trees : K3 minor-free graphs
Planar graphs : K5, K3,3 minor-free graphs (Kuratowski’s theorem)
Series-parallel graphs : K4 minor-free graphs
Outerplanar graphs : K4, K2,3 minor-free graphs
· · ·

SLIDE 7

Questions

How many of planar structures are there ?

(exactly / asymptotically)

SLIDE 8

Questions

How many of planar structures are there ?

(exactly / asymptotically)

What properties does a random planar structure have ?

SLIDE 9

Questions

How many of planar structures are there ?

(exactly / asymptotically)

What properties does a random planar structure have ?
what is the probability of being connected?
what is the chromatic number?

SLIDE 10

Questions

How many of planar structures are there ?

(exactly / asymptotically)

What properties does a random planar structure have ?
what is the probability of being connected?
what is the chromatic number?
How can we efficiently sample a random instance uniformly at

random?

SLIDE 11

Questions

How many of planar structures are there ?

(exactly / asymptotically)

What properties does a random planar structure have ?
what is the probability of being connected?
what is the chromatic number?
How can we efficiently sample a random instance uniformly at

random?

average case analysis
empirical properties

SLIDE 12

Scheme

Decomposition

SLIDE 13

Scheme

Decomposition Recursive Counting Formulas

SLIDE 14

Scheme

Decomposition Recursive Counting Formulas Uniform Generation Recursive Method

SLIDE 15

Scheme

Decomposition Equations of Generating Functions Recursive Counting Formulas Uniform Generation Recursive Method

SLIDE 16

Scheme

Singularity Analysis Decomposition Equations of Generating Functions Recursive Counting Formulas Uniform Generation Asymptotic Number Recursive Method

SLIDE 17

Scheme

Singularity Analysis Decomposition Equations of Generating Functions Recursive Counting Formulas Uniform Generation Asymptotic Number Typical Properties Recursive Method Probabilistic Analysis

SLIDE 18

Scheme

Singularity Analysis Decomposition Equations of Generating Functions Recursive Counting Formulas Uniform Generation Asymptotic Number Typical Properties Recursive Method Probabilistic Analysis Bolzmann Sampler

SLIDE 19

Labeled trees

How many labeled trees are there on vertex set [n] := {1, · · · , n}?

SLIDE 20

Labeled trees

How many labeled trees are there on vertex set [n] := {1, · · · , n}?

1

SLIDE 21

Labeled trees

How many labeled trees are there on vertex set [n] := {1, · · · , n}?

1

SLIDE 22

Labeled trees

How many labeled trees are there on vertex set [n] := {1, · · · , n}?

1 2

SLIDE 23

Labeled trees

How many labeled trees are there on vertex set [n] := {1, · · · , n}?

1 2

Let t(n) be the number of rooted trees on [n]

SLIDE 24

Labeled trees

How many labeled trees are there on vertex set [n] := {1, · · · , n}?

1 2

Let t(n) be the number of rooted trees on [n] t(n) n =

i

n − 2 i − 1

t(i)t(n − i)

n − i

SLIDE 25

Labeled trees

How many labeled trees are there on vertex set [n] := {1, · · · , n}?

t(i) t(n−i)

1 2

Let t(n) be the number of rooted trees on [n] t(n) n =

i

n − 2 i − 1

t(i)t(n − i)

n − i

SLIDE 26

Recursive method

[ NIJENHUIS, WILF 79; FLAJOLET, ZIMMERMAN, VAN CUTSEM 94 ]

t(n) n =

i

n − 2 i − 1

t(i)t(n − i)

n − i

SLIDE 27

Recursive method

[ NIJENHUIS, WILF 79; FLAJOLET, ZIMMERMAN, VAN CUTSEM 94 ]

t(n) n =

i

n − 2 i − 1

t(i)t(n − i)

n − i Uniform sampling algorithm for trees:

Generate(n): returns a random tree on [n]. choose a root vertex r with probability 1/n return Generate(n, r) Generate(n, r): returns a random tree on [n] with the root vertex r choose the order i of the split subtree with probability n `n−2

i−1

´ t(i)t(n − i)/((n − i)t(n)) let s = min([n] \ {r}) choose a random subset {s} ⊆ {w1, . . . , wi} ⊆ [n] \ {r} (with relative order) let {v1, . . . , vn−i} = [n] \ {w1, . . . , wi} (with relative order) T1 = Generate(i); relabel vertex j in T1 with wj (denote by r the root vertex of T1) T2 = Generate(n − i, r); relabel vertex j = r in T2 with vj return T1 ∪ T2 ∪ {(r, wr)} with marked r

SLIDE 28

Generating function

Let T(z) be the exponential generating function for rooted trees defined by T(z) =

n

t(n)zn n! .

SLIDE 29

Generating function

Let T(z) be the exponential generating function for rooted trees defined by T(z) =

n

t(n)zn n! . Then T(z)=z

1 + T(z) + T(z)2

2! +T(z)3 3! + · · ·

= zeT(z).

SLIDE 30

Generating function

Let T(z) be the exponential generating function for rooted trees defined by T(z) =

n

t(n)zn n! . Then T(z)=z

1 + T(z) + T(z)2

2! +T(z)3 3! + · · ·

= zeT(z).

SLIDE 31

Generating function

Let T(z) be the exponential generating function for rooted trees defined by T(z) =

n

t(n)zn n! . Then T(z)=z

1 + T(z) + T(z)2

2! +T(z)3 3! + · · ·

= zeT(z).

SLIDE 32

Exact number

LAGRANGE INVERSION THEOREM

[ FLAJOLET, SEDGEWICK 07+ ]

Let φ(u) =

k φkuk be a power series of C[[u]] with φ0 = 0. Then the

equation y = zφ(y) admits a unique solution in C[[z]] whose coefficients are given by y(z) =

n

ynzn where yn = 1 n[un−1]φ(u)n .

SLIDE 33

Exact number

LAGRANGE INVERSION THEOREM

[ FLAJOLET, SEDGEWICK 07+ ]

Let φ(u) =

k φkuk be a power series of C[[u]] with φ0 = 0. Then the

equation y = zφ(y) admits a unique solution in C[[z]] whose coefficients are given by y(z) =

n

ynzn where yn = 1 n[un−1]φ(u)n . From T(z) = zφ(T(z)) with φ(u) = eu, we have t(n) n! = 1 n[un−1]eun = 1 n[un−1]

k≥0

(un)k k! = 1 n nn−1 (n − 1)! = nn−1 n! .

SLIDE 34

Exact number

LAGRANGE INVERSION THEOREM

[ FLAJOLET, SEDGEWICK 07+ ]

Let φ(u) =

k φkuk be a power series of C[[u]] with φ0 = 0. Then the

equation y = zφ(y) admits a unique solution in C[[z]] whose coefficients are given by y(z) =

n

ynzn where yn = 1 n[un−1]φ(u)n . From T(z) = zφ(T(z)) with φ(u) = eu, we have t(n) n! = 1 n[un−1]eun = 1 n[un−1]

k≥0

(un)k k! = 1 n nn−1 (n − 1)! = nn−1 n! .

SLIDE 35

Exact number

LAGRANGE INVERSION THEOREM

[ FLAJOLET, SEDGEWICK 07+ ]

Let φ(u) =

k φkuk be a power series of C[[u]] with φ0 = 0. Then the

equation y = zφ(y) admits a unique solution in C[[z]] whose coefficients are given by y(z) =

n

ynzn where yn = 1 n[un−1]φ(u)n . From T(z) = zφ(T(z)) with φ(u) = eu, we have t(n) n! = 1 n[un−1]eun = 1 n[un−1]

k≥0

(un)k k! = 1 n nn−1 (n − 1)! = nn−1 n! . Thus the number of labeled trees on n vertices equals t(n)

n

= nn−2.

SLIDE 36

Asymptotic number

View a generating function T(z) =

n t(n)zn n! as a complex-valued

function that is analytic at the origin.

SLIDE 37

Asymptotic number

View a generating function T(z) =

n t(n)zn n! as a complex-valued

function that is analytic at the origin. Let R be the radius of convergence of T(z). Then [zn]T(z) = θ(n)R−n, where lim sup

n→∞ |θ(n)|1/n = 1.

SLIDE 38

Asymptotic number

View a generating function T(z) =

n t(n)zn n! as a complex-valued

function that is analytic at the origin. Let R be the radius of convergence of T(z). Then [zn]T(z) = θ(n)R−n, where lim sup

n→∞ |θ(n)|1/n = 1.

[Pringsheim’s Theorem] The point z = R is a dominant singularity of T(z), since T(z) has non-negative Taylor coefficients.

SLIDE 39

Asymptotic number

View a generating function T(z) =

n t(n)zn n! as a complex-valued

function that is analytic at the origin. Let R be the radius of convergence of T(z). Then [zn]T(z) = θ(n)R−n, where lim sup

n→∞ |θ(n)|1/n = 1.

[Pringsheim’s Theorem] The point z = R is a dominant singularity of T(z), since T(z) has non-negative Taylor coefficients. How to determine

the dominant singularity R and
the subexponential factor θ(n)?

SLIDE 40

Singularity analysis

[ FLAJOLET, SEDGEWICK 07+ ]

Let ψ(u) be the functional inverse of T(z). (Indeed ψ(u) = ue−u for rooted trees.)

SLIDE 41

Singularity analysis

[ FLAJOLET, SEDGEWICK 07+ ]

Let ψ(u) be the functional inverse of T(z). (Indeed ψ(u) = ue−u for rooted trees.) Let r > 0 be the radius of convergence of ψ, and suppose there exists u0 ∈ (0, r) such that ψ(u0) = 0 and ψ(u0) = 0.

u0 ψ(u)

Indeed, z0 = e−1 and thus

t(n) = θ(n)en, where lim sup

|θ(n)|1/n = 1.

SLIDE 42

Singularity analysis

[ FLAJOLET, SEDGEWICK 07+ ]

Let ψ(u) be the functional inverse of T(z). (Indeed ψ(u) = ue−u for rooted trees.) Let r > 0 be the radius of convergence of ψ, and suppose there exists u0 ∈ (0, r) such that ψ(u0) = 0 and ψ(u0) = 0.

u0 z0 = ψ(u0) T(z) ψ(u)

Indeed, z0 = e−1 and thus

t(n) = θ(n)en, where lim sup

|θ(n)|1/n = 1.

SLIDE 43

Singularity analysis

[ FLAJOLET, SEDGEWICK 07+ ]

Let ψ(u) be the functional inverse of T(z). (Indeed ψ(u) = ue−u for rooted trees.) Let r > 0 be the radius of convergence of ψ, and suppose there exists u0 ∈ (0, r) such that ψ(u0) = 0 and ψ(u0) = 0.

u0 z0 = ψ(u0) T(z) ψ(u)

Indeed, z0 = e−1 and thus

t(n) = θ(n)en, where lim sup

|θ(n)|1/n = 1.

SLIDE 44

Singularity analysis

[ FLAJOLET, SEDGEWICK 07+ ]

Let ψ(u) be the functional inverse of T(z). (Indeed ψ(u) = ue−u for rooted trees.) Let r > 0 be the radius of convergence of ψ, and suppose there exists u0 ∈ (0, r) such that ψ(u0) = 0 and ψ(u0) = 0.

u0 z0 = ψ(u0) T(z) ψ(u)

Indeed, z0 = e−1 and thus t(n)

n! = θ(n)en, where lim sup |θ(n)|1/n = 1.

SLIDE 45

Local dependency

[ FLAJOLET, SEDGEWICK 07+ ]

Taylor expansion of z = ψ(u) at u0 is of the form ψ(u) = ψ(u0) + 1 2ψ(u0)(u − u0)2 + · · · .

SLIDE 46

Local dependency

[ FLAJOLET, SEDGEWICK 07+ ]

Taylor expansion of z = ψ(u) at u0 is of the form ψ(u) = ψ(u0) + 1 2ψ(u0)(u − u0)2 + · · · . It implies a locally quadratic dependency between z and u = T(z): (T(z) − T(z0))2 = (u − u0)2 ∼ 2 ψ(u0)(z − z0) = −2ψ(u0) ψ(u0)(1 − z/z0)

SLIDE 47

Local dependency

[ FLAJOLET, SEDGEWICK 07+ ]

Taylor expansion of z = ψ(u) at u0 is of the form ψ(u) = ψ(u0) + 1 2ψ(u0)(u − u0)2 + · · · . It implies a locally quadratic dependency between z and u = T(z): (T(z) − T(z0))2 = (u − u0)2 ∼ 2 ψ(u0)(z − z0) = −2ψ(u0) ψ(u0)(1 − z/z0)

SLIDE 48

Local dependency

[ FLAJOLET, SEDGEWICK 07+ ]

Taylor expansion of z = ψ(u) at u0 is of the form ψ(u) = ψ(u0) + 1 2ψ(u0)(u − u0)2 + · · · . It implies a locally quadratic dependency between z and u = T(z): (T(z) − T(z0))2 = (u − u0)2 ∼ 2 ψ(u0)(z − z0) = −2ψ(u0) ψ(u0)(1 − z/z0) Since T(z) is increasing along the positive real axis, we have T(z) − T(z0) ∼ −

−2ψ(u0)/ψ(u0) (1 − z/z0)1/2

SLIDE 49

Local dependency

[ FLAJOLET, SEDGEWICK 07+ ]

Taylor expansion of z = ψ(u) at u0 is of the form ψ(u) = ψ(u0) + 1 2ψ(u0)(u − u0)2 + · · · . It implies a locally quadratic dependency between z and u = T(z): (T(z) − T(z0))2 = (u − u0)2 ∼ 2 ψ(u0)(z − z0) = −2ψ(u0) ψ(u0)(1 − z/z0) Since T(z) is increasing along the positive real axis, we have T(z) − T(z0) ∼ −

−2ψ(u0)/ψ(u0) (1 − z/z0)1/2

Using ∆-analycity of T(z) and transfer theorem, we have [zn]T(z) ∼ −

−2ψ(u0)/ψ(u0)[zn](1 − z/z0)1/2

SLIDE 50

Basic scale

[ FLAJOLET, SEDGEWICK 07+ ]

We have z0 = e−1, u0 = 1, ψ(u) = ue−u and [zn]T(z) ∼ −

−2ψ(u0)/ψ(u0)[zn](1 − z/z0)1/2.

SLIDE 51

Basic scale

[ FLAJOLET, SEDGEWICK 07+ ]

We have z0 = e−1, u0 = 1, ψ(u) = ue−u and [zn]T(z) ∼ −

−2ψ(u0)/ψ(u0)[zn](1 − z/z0)1/2.

RESCALING RULE/ GENERALIZED BINOMIAL THEOREM

[zn] (1 − z/z0)1/2 = n − 3/2 n

z−n

∼ n−3/2 −2√πz−n

0 .

SLIDE 52

Basic scale

[ FLAJOLET, SEDGEWICK 07+ ]

We have z0 = e−1, u0 = 1, ψ(u) = ue−u and [zn]T(z) ∼ −

−2ψ(u0)/ψ(u0)[zn](1 − z/z0)1/2.

RESCALING RULE/ GENERALIZED BINOMIAL THEOREM

[zn] (1 − z/z0)1/2 = n − 3/2 n

z−n

∼ n−3/2 −2√πz−n

0 .

We have that the number of rooted trees on n vertices equals nn−1 = t(n) ∼ 1 √ 2πn−3/2enn! ∼ 1 √ 2πn−3/2enn e n √ 2πn (Stirling’s formula) = nn−1 (Cayley’s formula)

SLIDE 53

Basic scale

[ FLAJOLET, SEDGEWICK 07+ ]

We have z0 = e−1, u0 = 1, ψ(u) = ue−u and [zn]T(z) ∼ −

−2ψ(u0)/ψ(u0)[zn](1 − z/z0)1/2.

RESCALING RULE/ GENERALIZED BINOMIAL THEOREM

[zn] (1 − z/z0)1/2 = n − 3/2 n

z−n

∼ n−3/2 −2√πz−n

0 .

We have that the number of rooted trees on n vertices equals nn−1 = t(n) ∼ 1 √ 2πn−3/2enn! ∼ 1 √ 2πn−3/2enn e n √ 2πn (Stirling’s formula) = nn−1 (Cayley’s formula)

SLIDE 54

Basic scale

[ FLAJOLET, SEDGEWICK 07+ ]

We have z0 = e−1, u0 = 1, ψ(u) = ue−u and [zn]T(z) ∼ −

−2ψ(u0)/ψ(u0)[zn](1 − z/z0)1/2.

RESCALING RULE/ GENERALIZED BINOMIAL THEOREM

[zn] (1 − z/z0)1/2 = n − 3/2 n

z−n

∼ n−3/2 −2√πz−n

0 .

We have that the number of rooted trees on n vertices equals nn−1 = t(n) ∼ 1 √ 2πn−3/2enn! ∼ 1 √ 2πn−3/2enn e n √ 2πn (Stirling’s formula) = nn−1 (Cayley’s formula)

SLIDE 55

Block structure of a graph

A block of a graph is a maximal connected subgraph without a cutvertex:

SLIDE 56

Block structure of a graph

A block of a graph is a maximal connected subgraph without a cutvertex:

a maximal biconnected subgraph,
an edge (including its ends), or
an isolated vertex

SLIDE 57

Block structure of a graph

A block of a graph is a maximal connected subgraph without a cutvertex:

a maximal biconnected subgraph,
an edge (including its ends), or
an isolated vertex

The block structure of a graph is a forest with two types of vertices: the blocks and the cutvertices of the graph.

SLIDE 58

Blocks of planar structures

2-connected outerplanar graphs:

SLIDE 59

Blocks of planar structures

2-connected outerplanar graphs:

[ BODIRSKY, GIMÉNEZ, K., NOY 07+ ]

# outerplanar graphs on n vertices ∼ α n−5/2 ρn n! , ρ . = 7.32

SLIDE 60

Blocks of planar structures

2-connected outerplanar graphs:

[ BODIRSKY, GIMÉNEZ, K., NOY 07+ ]

# outerplanar graphs on n vertices ∼ α n−5/2 ρn n! , ρ . = 7.32 2-connected series-parallel graphs:

SLIDE 61

Blocks of planar structures

2-connected outerplanar graphs:

[ BODIRSKY, GIMÉNEZ, K., NOY 07+ ]

# outerplanar graphs on n vertices ∼ α n−5/2 ρn n! , ρ . = 7.32 2-connected series-parallel graphs:

SLIDE 62

Blocks of planar structures

2-connected outerplanar graphs:

[ BODIRSKY, GIMÉNEZ, K., NOY 07+ ]

# outerplanar graphs on n vertices ∼ α n−5/2 ρn n! , ρ . = 7.32 2-connected series-parallel graphs:

SLIDE 63

Blocks of planar structures

2-connected outerplanar graphs:

[ BODIRSKY, GIMÉNEZ, K., NOY 07+ ]

# outerplanar graphs on n vertices ∼ α n−5/2 ρn n! , ρ . = 7.32 2-connected series-parallel graphs:

[ BODIRSKY, GIMÉNEZ, K., NOY 07+ ]

# series-parallel graphs on n vertices ∼ β n−5/2 γn n! , γ . = 9.07

SLIDE 64

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ]

SLIDE 65

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ]

SLIDE 66

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ]

SLIDE 67

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ]

SLIDE 68

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ] [ BENDER, GAO, WORMALD 02 ]

The growth constant for biconnected planar graphs: ∼ 26.18

SLIDE 69

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ] [ BENDER, GAO, WORMALD 02 ]

The growth constant for biconnected planar graphs: ∼ 26.18

[ BENDER, RICHMOND 84 ; BODIRSKY, GRÖPL, JOHANNSEN, K. 05; FUSY, POULALHON, SCHAEFFER 05 ]

The growth constant for 3-connected planar graphs: ∼ 21.05

SLIDE 70

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ] [ BENDER, GAO, WORMALD 02 ]

The growth constant for biconnected planar graphs: ∼ 26.18

[ BENDER, RICHMOND 84 ; BODIRSKY, GRÖPL, JOHANNSEN, K. 05; FUSY, POULALHON, SCHAEFFER 05 ]

The growth constant for 3-connected planar graphs: ∼ 21.05 Uniform sampling algorithm

SLIDE 71

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ] [ BENDER, GAO, WORMALD 02 ]

The growth constant for biconnected planar graphs: ∼ 26.18

[ BENDER, RICHMOND 84 ; BODIRSKY, GRÖPL, JOHANNSEN, K. 05; FUSY, POULALHON, SCHAEFFER 05 ]

The growth constant for 3-connected planar graphs: ∼ 21.05 Uniform sampling algorithm

SLIDE 72

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ] [ BENDER, GAO, WORMALD 02 ]

The growth constant for biconnected planar graphs: ∼ 26.18

[ BENDER, RICHMOND 84 ; BODIRSKY, GRÖPL, JOHANNSEN, K. 05; FUSY, POULALHON, SCHAEFFER 05 ]

The growth constant for 3-connected planar graphs: ∼ 21.05 Uniform sampling algorithm

[ BODIRSKY, GRÖPL, K. 03; FUSY 05 ; GIMÉNEZ, NOY 05 ]

Uniform sampling algorithm for planar graphs O(n7); O(n2) The number of planar graphs is ∼ c n−7/2 27.22nn!

SLIDE 73

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ] [ BENDER, GAO, WORMALD 02 ]

The growth constant for biconnected planar graphs: ∼ 26.18

[ BENDER, RICHMOND 84 ; BODIRSKY, GRÖPL, JOHANNSEN, K. 05; FUSY, POULALHON, SCHAEFFER 05 ]

The growth constant for 3-connected planar graphs: ∼ 21.05 Uniform sampling algorithm

[ BODIRSKY, GRÖPL, K. 03; FUSY 05 ; GIMÉNEZ, NOY 05 ]

Uniform sampling algorithm for planar graphs O(n7); O(n2) The number of planar graphs is ∼ c n−7/2 27.22nn!

SLIDE 74

Labeled planar graphs

2-connected graphs

[ TRAKHTENBROT 58; TUTTE 63; WALSH 82 ] [ BENDER, GAO, WORMALD 02 ]

The growth constant for biconnected planar graphs: ∼ 26.18

[ BENDER, RICHMOND 84 ; BODIRSKY, GRÖPL, JOHANNSEN, K. 05; FUSY, POULALHON, SCHAEFFER 05 ]

The growth constant for 3-connected planar graphs: ∼ 21.05 Uniform sampling algorithm

[ BODIRSKY, GRÖPL, K. 03; FUSY 05 ; GIMÉNEZ, NOY 05 ]

Uniform sampling algorithm for planar graphs O(n7); O(n2) The number of planar graphs is ∼ c n−7/2 27.22nn!

SLIDE 75

Scheme

Singularity Analysis Decomposition Equations of Generating Functions Recursive Counting Formulas Uniform Generation Asymptotic Number Recursive Method

SLIDE 76

Scheme

Singularity Analysis Decomposition Equations of Generating Functions Recursive Counting Formulas Uniform Generation Asymptotic Number Typical Properties Recursive Method Probabilistic Analysis

SLIDE 77

Labeled cubic planar graphs

[ BODIRSKY, K., LÖFFLER, MCDIARMID 07 ]

The number of cubic planar graphs on n vertices is asymptotically ∼ αn−7/2ρnn! , where ρ . = 3.1325

SLIDE 78

Labeled cubic planar graphs

[ BODIRSKY, K., LÖFFLER, MCDIARMID 07 ]

The number of cubic planar graphs on n vertices is asymptotically ∼ αn−7/2ρnn! , where ρ . = 3.1325 What is the chromatic number of a random cubic planar graph G that is chosen uniformly at random among labeled cubic planar graphs on [n]?

SLIDE 79

Chromatic number

What is the chromatic number of a random cubic planar graph G?

χ(G) ≤ 4

[Four colour theorem]

For any connected graph G that is neither a complete graph nor an
dd cycle, χ(G) ≤ ∆(G) = 3

[Brooks’ theorem]

SLIDE 80

Chromatic number

What is the chromatic number of a random cubic planar graph G?

χ(G) ≤ 4

[Four colour theorem]

For any connected graph G that is neither a complete graph nor an
dd cycle, χ(G) ≤ ∆(G) = 3

[Brooks’ theorem] If G contains a component isomorphic to K4, then χ(G) = 4. If G contains no isolated K4, but at least one triangle, then χ(G) = 3.

SLIDE 81

Random cubic planar graphs

[ BODIRSKY, K., LÖFFLER, MCDIARMID 07 ]

Let G(k)

n

be a random k connected cubic planar graph on n vertices.

SLIDE 82

Random cubic planar graphs

[ BODIRSKY, K., LÖFFLER, MCDIARMID 07 ]

Let G(k)

n

be a random k connected cubic planar graph on n vertices.

SUBGRAPH CONTAINMENTS

Let Xn be # isolated K4’s in G(0)

n

and Yn # triangles in G(k)

n , k > 0. Then

lim

n→∞ Pr(Xn > 0) = 1 − e− ρ4

4! ,

lim

n→∞ Pr(Yn > 0) = 1.

SLIDE 83

Random cubic planar graphs

[ BODIRSKY, K., LÖFFLER, MCDIARMID 07 ]

Let G(k)

n

be a random k connected cubic planar graph on n vertices.

SUBGRAPH CONTAINMENTS

Let Xn be # isolated K4’s in G(0)

n

and Yn # triangles in G(k)

n , k > 0. Then

lim

n→∞ Pr(Xn > 0) = 1 − e− ρ4

4! ,

lim

n→∞ Pr(Yn > 0) = 1.

CHROMATIC NUMBER

lim

n→∞ Pr(χ(G(0) n ) = 4) = lim n→∞ Pr(Xn > 0) = 1 − e− ρ4

4!

lim

n→∞ Pr(χ(G(0) n ) = 3) = lim n→∞ Pr(Xn = 0, Yn > 0) = e− ρ4

4! .

= 0.9995 . For k = 1, 2, 3, limn→∞ Pr(χ(G(k)

n ) = 3) = limn→∞ Pr(Yn > 0) = 1 .

SLIDE 84

Labeled planar structures

The number of planar structures on n vertices is asymp. ∼ α n−β γnn!. Let Gn be a random planar structure on n vertices. Then as n → ∞,

the expected number of edges in Gn is

∼ µn,

Gn is connected with probability tending to a constant pcon, and
χ(Gn) is three with probability tending to a constant pχ.

Running time of uniform sampler (recursive method): ˜ O(nk)

Classes β γ µ pcon pχ k Trees 5/2 2.71 1 1 4 Outerplanar graphs 5/2 7.32 1.56 0.861 1 4 Series-parallel graphs 5/2 9.07 1.61 0.889 ? ? Planar graphs 7/2 27.2 2.21 0.963 ? 7 Cubic planar graphs 7/2 3.13 1.50 ≥ 0.998 0.999 6

SLIDE 85

Labeled planar structures

The number of planar structures on n vertices is asymp. ∼ α n−β γnn!. Let Gn be a random planar structure on n vertices. Then as n → ∞,

the expected number of edges in Gn is

∼ µn,

Gn is connected with probability tending to a constant pcon, and
χ(Gn) is three with probability tending to a constant pχ.

Running time of uniform sampler (recursive method): ˜ O(nk)

Classes β γ µ pcon pχ k Trees 5/2 2.71 1 1 4 Outerplanar graphs 5/2 7.32 1.56 0.861 1 4 Series-parallel graphs 5/2 9.07 1.61 0.889 ? ? Planar graphs 7/2 27.2 2.21 0.963 ? 7 Cubic planar graphs 7/2 3.13 1.50 ≥ 0.998 0.999 6

SLIDE 86

Labeled planar structures

The number of planar structures on n vertices is asymp. ∼ α n−β γnn!. Let Gn be a random planar structure on n vertices. Then as n → ∞,

the expected number of edges in Gn is

∼ µn,

Gn is connected with probability tending to a constant pcon, and
χ(Gn) is three with probability tending to a constant pχ.

Running time of uniform sampler (recursive method): ˜ O(nk)

Classes β γ µ pcon pχ k Trees 5/2 2.71 1 1 4 Outerplanar graphs 5/2 7.32 1.56 0.861 1 4 Series-parallel graphs 5/2 9.07 1.61 0.889 ? ? Planar graphs 7/2 27.2 2.21 0.963 ? 7 Cubic planar graphs 7/2 3.13 1.50 ≥ 0.998 0.999 6

SLIDE 87

Unlabeled planar structures

Difficulty with unlabeled planar structures is symmetry:

SLIDE 88

Unlabeled planar structures

Difficulty with unlabeled planar structures is symmetry:

recursive method: decomposition along symmetry
Pólya Theory: cycle indices

SLIDE 89

Unlabeled planar structures

Difficulty with unlabeled planar structures is symmetry:

recursive method: decomposition along symmetry
Pólya theory: symmetry vs orbits of automorphism group of a graph
Boltzmann sampler: composition operation, cycle-pointing

Bodirsky, K. 06

Uniform sampling Outerplanar graphs Asymptotic number

Bodirsky, Groepl, K. 04+

Cubic planar graphs

Bodirsky, Groepl, K. 05

2−con planar graphs Planar graphs

SLIDE 90

Unlabeled planar structures

Difficulty with unlabeled planar structures is symmetry:

recursive method: decomposition along symmetry
Pólya theory: symmetry vs orbits of automorphism group of a graph
Boltzmann sampler: composition operation, cycle-pointing

Bodirsky, K. 06

Uniform sampling Outerplanar graphs Asymptotic number

cn^{−5/2}7.5^n

Bodirsky, Fusy, K., Vigerske 07+ Bodirsky, Groepl, K. 04+

Cubic planar graphs

Bodirsky, Groepl, K. 05

2−con planar graphs Planar graphs

SLIDE 91

Unlabeled planar structures

Difficulty with unlabeled planar structures is symmetry:

recursive method: decomposition along symmetry
Pólya theory: symmetry vs orbits of automorphism group of a graph
Boltzmann sampler: composition operation, cycle-pointing

Bodirsky, K. 06

Uniform sampling Outerplanar graphs Asymptotic number

cn^{−5/2}7.5^n

Bodirsky, Fusy, K., Vigerske 07+ Bodirsky, Fusy, K., Vigerske 07 Bodirsky, Groepl, K. 04+

Cubic planar graphs

Bodirsky, Groepl, K. 05

2−con planar graphs Planar graphs

SLIDE 92

Unlabeled planar structures

Difficulty with unlabeled planar structures is symmetry:

recursive method: decomposition along symmetry
Pólya theory: symmetry vs orbits of automorphism group of a graph
Boltzmann sampler: composition operation, cycle-pointing

Bodirsky, K. 06

Uniform sampling Outerplanar graphs Asymptotic number

cn^{−5/2}7.5^n

Bodirsky, Fusy, K., Vigerske 07+ Bodirsky, Fusy, K., Vigerske 07 Bodirsky, Groepl, K. 04+

Cubic planar graphs

?

Bodirsky, Groepl, K. 05

?

2−con planar graphs Planar graphs

? ?

SLIDE 93

Outline

Decomposition
Recursive method
Singularity analysis
Probabilistic analysis

SLIDE 94

Outline

Decomposition
Recursive method
Singularity analysis
Probabilistic analysis
Gaussian matrix integral

SLIDE 95

Gaussian matrix integral

[ WICK 50 ]

Let M = (Mij) be an N × N Hermitian matrix (i.e., Mij = Mji) and dM =

i dMii

i<j d Re(Mij)d Im(Mij) the standard Haar measure.

SLIDE 96

Gaussian matrix integral

[ WICK 50 ]

Let M = (Mij) be an N × N Hermitian matrix (i.e., Mij = Mji) and dM =

i dMii

i<j d Re(Mij)d Im(Mij) the standard Haar measure.

SLIDE 97

Gaussian matrix integral

[ WICK 50 ]

Let M = (Mij) be an N × N Hermitian matrix (i.e., Mij = Mji) and dM =

i dMii

i<j d Re(Mij)d Im(Mij) the standard Haar measure.

The Gaussian matrix integral is defined by < f > =

f(M)e−N Tr( M2

2 )dM

e−N Tr( M2

2 )dM

, where the integration is over N × N Hermitian matrices.

SLIDE 98

Gaussian matrix integral

[ WICK 50 ]

Let M = (Mij) be an N × N Hermitian matrix (i.e., Mij = Mji) and dM =

i dMii

i<j d Re(Mij)d Im(Mij) the standard Haar measure.

The Gaussian matrix integral is defined by < f > =

f(M)e−N Tr( M2

2 )dM

e−N Tr( M2

2 )dM

, where the integration is over N × N Hermitian matrices. Using the source integral < eTr(MS) >, we obtain

< MijMkl > = ∂ ∂Sji ∂ ∂Slk < eTr(MS) > ˛ ˛ ˛

S=0=

∂ ∂Sji ∂ ∂Slk e

Tr(S2) 2N

˛ ˛ ˛

S=0= δilδjk

N .

SLIDE 99

Gaussian matrix integral

[ WICK 50 ]

Let M = (Mij) be an N × N Hermitian matrix (i.e., Mij = Mji) and dM =

i dMii

i<j d Re(Mij)d Im(Mij) the standard Haar measure.

The Gaussian matrix integral is defined by < f > =

f(M)e−N Tr( M2

2 )dM

e−N Tr( M2

2 )dM

, where the integration is over N × N Hermitian matrices. Using the source integral < eTr(MS) >, we obtain

< MijMkl > = ∂ ∂Sji ∂ ∂Slk < eTr(MS) > ˛ ˛ ˛

S=0=

∂ ∂Sji ∂ ∂Slk e

Tr(S2) 2N

˛ ˛ ˛

S=0= δilδjk

N .

SLIDE 100

Pictorial interpretation

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04]

Pictorial interpretation from < MijMkl > =

δilδjk N

:

i j

Mij

i j l, k, l = i k = j

< MijMkl >=

1 N

SLIDE 101

Pictorial interpretation

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04]

Pictorial interpretation from < MijMkl > =

δilδjk N

:

i j

Mij

i j l, k, l = i k = j

< MijMkl >=

1 N

<Tr(Mn)> = < X

1≤i1,i2,··· ,in≤N

Mi1i2Mi2i3 · · · Mini1> = X

1≤i1,i2,··· ,in≤N

X

P

Y

(ikik+1,ilil+1)∈P

δikik+1δilil+1 N where P is a partition of {i1i2, i2i3, · · · , ini1} into pairs.

SLIDE 102

Pictorial interpretation

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04]

Pictorial interpretation from < MijMkl > =

δilδjk N

:

i j

Mij

i j l, k, l = i k = j

< MijMkl >=

1 N

<Tr(Mn)> = < X

1≤i1,i2,··· ,in≤N

Mi1i2Mi2i3 · · · Mini1> = X

1≤i1,i2,··· ,in≤N

X

P

Y

(ikik+1,ilil+1)∈P

δikik+1δilil+1 N where P is a partition of {i1i2, i2i3, · · · , ini1} into pairs.

i3 in i1i2 Mi1i2 Mi2i3 · · · Mini1 > <

SLIDE 103

Pictorial interpretation

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04]

Pictorial interpretation from < MijMkl > =

δilδjk N

:

i j

Mij

i j l, k, l = i k = j

< MijMkl >=

1 N

<Tr(Mn)> = < X

1≤i1,i2,··· ,in≤N

Mi1i2Mi2i3 · · · Mini1> = X

1≤i1,i2,··· ,in≤N

X

P

Y

(ikik+1,ilil+1)∈P

δikik+1δilil+1 N where P is a partition of {i1i2, i2i3, · · · , ini1} into pairs.

i3 in i1i2 Mi1i2 Mi2i3 · · · Mini1 > <

SLIDE 104

Fat graphs

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04] < Tr(Mn) > = X

1≤i1,i2,··· ,in≤N

X

P

Y

(ikik+1,ilil+1)∈P

δikik+1δilil+1 N .

SLIDE 105

Fat graphs

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04] < Tr(Mn) > = X

1≤i1,i2,··· ,in≤N

X

P

Y

(ikik+1,ilil+1)∈P

δikik+1δilil+1 N .

A pairing P with non-zero contribution to < Tr(Mn) > ⇐ ⇒ a fat graph with one island and n/2 fat edges ordered cyclically. (It defines uniquely an embedding on a surface: a map!)

SLIDE 106

Fat graphs

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04] < Tr(Mn) > = X

1≤i1,i2,··· ,in≤N

X

P

Y

(ikik+1,ilil+1)∈P

δikik+1δilil+1 N .

A pairing P with non-zero contribution to < Tr(Mn) > ⇐ ⇒ a fat graph with one island and n/2 fat edges ordered cyclically. (It defines uniquely an embedding on a surface: a map!)

SLIDE 107

Fat graphs

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04] < Tr(Mn) > = X

1≤i1,i2,··· ,in≤N

X

P

Y

(ikik+1,ilil+1)∈P

δikik+1δilil+1 N .

Let F be a fat graph with one island, e(F) edges and f(F) faces.

The edges contribute N −e(F), since each edge contributes N −1.
The faces contribute Nf(F), since each face attains independently

any index from 1 to N.

i1

SLIDE 108

Fat graphs

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04] < Tr(Mn) > = X

1≤i1,i2,··· ,in≤N

X

P

Y

(ikik+1,ilil+1)∈P

δikik+1δilil+1 N .

Let F be a fat graph with one island, e(F) edges and f(F) faces.

The edges contribute N −e(F), since each edge contributes N −1.
The faces contribute Nf(F), since each face attains independently

any index from 1 to N. Thus < Tr(Mn) > =

F

N−e(F)+f(F) where the sum is over all fat graphs F with one island.

SLIDE 109

Fat graphs

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04]

Similarly we obtain <

NTr(M3)

4 NTr(M2) 3 > =

F

N7−e(F)+f(F), where the sum is over all fat graphs F with four islands of degree 3, and three islands of degree 2.

SLIDE 110

Fat graphs

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04]

Similarly we obtain <

NTr(M3)

4 NTr(M2) 3 > =

F

N7−e(F)+f(F), where the sum is over all fat graphs F with four islands of degree 3, and three islands of degree 2. An example of such a fat graph (i.e., a map)

SLIDE 111

Fat graphs

[ BRÉZIN, ITZYKSON, PARISI, ZUBER 78; ZVONKIN 97; DI FRANCESCO 04]

Similarly we obtain <

NTr(M3)

4 NTr(M2) 3 > =

F

N7−e(F)+f(F), where the sum is over all fat graphs F with four islands of degree 3, and three islands of degree 2. An example of such a fat graph (i.e., a map)

SLIDE 112

Planar maps

[ BOUTTIER, DI FRANCESCO, GUITTER 02 ]

Let g(M) = e

P

i≥1 zi i [NTr(Mi)]. Then

< g > =

a=(n1,··· ,nk)
F

Nv(F)−e(F)+f(F)

i≤k

zni

i

inini!, where F is a map with ni vertices of degree i. Furthermore, lim

N→∞

log < g > N2 =

a=(n1,··· ,nk)
Fc

N−2g(Fc)

i≤k

zni

i

inini! where Fcp is a connected map with ni vertices of degree i.

SLIDE 113

Planar maps

[ BOUTTIER, DI FRANCESCO, GUITTER 02 ]

Let g(M) = e

P

i≥1 zi i [NTr(Mi)]. Then

< g > =

a=(n1,··· ,nk)
F

Nv(F)−e(F)+f(F)

i≤k

zni

i

inini!, where F is a map with ni vertices of degree i. Furthermore, lim

N→∞

log < g > N2 = lim

N→∞

a=(n1,··· ,nk)
F cp

N−2g(Fc)

i≤k

zni

i

inini! where F cp is a connected planar map with ni vertices of degree i.

SLIDE 114

Planar maps

[ BOUTTIER, DI FRANCESCO, GUITTER 02 ]

Let g(M) = e

P

i≥1 zi i [NTr(Mi)]. Then

< g > =

a=(n1,··· ,nk)
F

Nv(F)−e(F)+f(F)

i≤k

zni

i

inini!, where F is a map with ni vertices of degree i. Furthermore, lim

N→∞

log < g > N2 = lim

N→∞

a=(n1,··· ,nk)
F cp

N−2g(Fc)

i≤k

zni

i

inini! where F cp is a connected planar map with ni vertices of degree i.

[ K., LOEBL 06+ ]

The number of planar graphs with a given degree sequence can also be formulated by a Gaussian matrix intergral.

SLIDE 115

Concluding remarks

Relevant work

There exists a constant c such that the number of graphs in a proper

minor-closed class ≤ cn n!

[ NORINE, SEYMOUR, THOMAS, WOLLAN 06 ]

Asymptotic growth of minor-closed classes of graphs

[ BERNARDI, NOY, WELSH 07+ ]

SLIDE 116

Concluding remarks

Relevant work

There exists a constant c such that the number of graphs in a proper

minor-closed class ≤ cn n!

[ NORINE, SEYMOUR, THOMAS, WOLLAN 06 ]

Asymptotic growth of minor-closed classes of graphs

[ BERNARDI, NOY, WELSH 07+ ]

SLIDE 117

Concluding remarks

Relevant work

There exists a constant c such that the number of graphs in a proper

minor-closed class ≤ cn n!

[ NORINE, SEYMOUR, THOMAS, WOLLAN 06 ]

Asymptotic growth of minor-closed classes of graphs

[ BERNARDI, NOY, WELSH 07+ ]

Open problems What are the asymptotic numbers of (1) unlabeled planar graphs (2) planar graphs with a given degree sequence (3) embeddable graphs on a surface with higer genus? What do random graphs chosen among (1), (2) or (3) look like? What structural properties of graphs determine the critical exponents of their asymptotic number?

SLIDE 118

Concluding remarks

Relevant work

There exists a constant c such that the number of graphs in a proper

minor-closed class ≤ cn n!

[ NORINE, SEYMOUR, THOMAS, WOLLAN 06 ]

Asymptotic growth of minor-closed classes of graphs

[ BERNARDI, NOY, WELSH 07+ ]

Open problems What are the asymptotic numbers of (1) unlabeled planar graphs (2) planar graphs with a given degree sequence (3) embeddable graphs on a surface with higer genus? What do random graphs chosen among (1), (2) or (3) look like? What structural properties of graphs determine the critical exponents of their asymptotic number?

SLIDE 119

Concluding remarks

Relevant work

There exists a constant c such that the number of graphs in a proper

minor-closed class ≤ cn n!

[ NORINE, SEYMOUR, THOMAS, WOLLAN 06 ]

Asymptotic growth of minor-closed classes of graphs

[ BERNARDI, NOY, WELSH 07+ ]

Open problems What are the asymptotic numbers of (1) unlabeled planar graphs (2) planar graphs with a given degree sequence (3) embeddable graphs on a surface with higer genus? What do random graphs chosen among (1), (2) or (3) look like? What structural properties of graphs determine the critical exponents of their asymptotic numbers?