Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles

8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles

1/73

Elementary

polycycles

Michel DezaENS, Paris, and ISM, Tokyo

Mathieu Dutour SikiricRudjer Boskovic Institute, Zagreb,

and ISM, Tokyo

and Mikhail Shtogrin

Steklov Institute, Moscow

p.1/4


2/73

I. -polycycles

p.2/4


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Definition

Given

and

, a

-polycycle is a non-empty

-connected plane, locally finite graph

with facespartitionned in two sets

and

(

is non-empty), so that:

all elements of

(called proper faces) arecombinatorial

-gons with

;

all elements of

(called holes) are pair-wisely disjoint,i.e. have no common vertices;

all vertices have degree within

and all interiorvertices are -valent.

p.3/4


4/73

Examples with one hole

A

!

"

-polycycle A

"

!

-polycycle

A

"

!

-polycycle A

"

#

-polycycle

p.4/4


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Examples with two hole or more

A

"

-polycycle A

!

"

-polycycle

p.5/4


6/73

$ -polycycles

Any

"

"

-polycycle is one of the following

Tetrahedron (with no hole):

"

following polycycles (with one hole):

p.6/4


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$ -polycycles

Any

"

-polycycle is one of the following

Cube (with no hole):

"

following polycycles (with one hole)

Following infinite family (with one hole):

p.7/4


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$ -polycycles

The infinite family%

&

( 0

1 (with two holes)

Following two infinite

"

-polycycles:

singly infinite polycycle doubly infinite polycycle

p.7/4


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$ -polycycles

Octahedron (with no hole):

Following polycycles (with one hole)

p.8/4


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$ -polycycles

Following infinite family (with one hole):

The infinite family3

%

&

( 0

1 (with two holes)

Following two infinite

"

-polycycles:

singly infinite polycycle doubly infinite polycycle p.8/4


11/73

Curvature conditions

A

-polycycle is called elliptic, parabolic or

hyperbolic if

5

6

789

@

A

B

C

D

is positive, zero or negative,

respectively.

Elliptic cases:

E

"

and

with F GH CI P

Q

!

E

and

with F GH CI P

Q

"

E

!

and

with F GH CI P

Q

"

Parabolic cases:

E

"

and

withF

G

H

C

I

P

E

#

E

and

with F GH CI P

E

E

#

and

with F GH CI P

E

#

All other cases are hyperbolic.

p.9/4


12/73

Limit case R S , S T

Elliptic

&

-polycycles:

!

Platonic solids

Tetra-

hedron

Cube

Bonjour

Octa-

hedron

Icosa-

hedron

Dodeca-

hedron

Parabolic

&

-polycycles:"

regular plane tilings

Hyperbolic

&

-polycycles: infinity

p.10/4


13/73

Generalization and T $ -polycycles

A generalization of

-polycycle is

U

-polycycles:the valency of interior vertices belong to a set

U

. All thetheory extends to this case.

A

&

-polycycle is a

&

-polycycle with only onehole (the exterior one). Their theory has additionalfeatures:

There exist a canonical model of them in the form of

&

5

regular partition.

For any

&

-polycycle%

, simple connectedness of%

ensures the existence of a canonical map from

%

to

&

5

.

p.11/4


14/73

Main examples of T $ -polycycles

Elliptic Parabolic Hyperbolic

&

"

"

"

"

all

!

"

"

!

"

#

#

"

othersExp. VW ,X

W , YW ,`

a ,b

c a

d

#

W

"

e

&

5

reg.part of spheref

of Euclidean of hyperbolic

plane

g

plane

h

Bonjour

domino diamond hexagon

Polyominoes: Conway, Penrose, Colomb (games, tilers ofg

, etc.), enumeration (in Physics, Statistical Mechanics).

Polyhexes: application in Organic Chemistry.

p.12/4


15/73

II. Decomposition

into elementary

polycycles

p.13/4

El l l


16/73

Elementary polycycles

A bridge of a

-polycycle is an edge, which is noton a boundary and goes from a hole to a hole (possibly,the same).

An elementary

-polycycle is one without bridges.

Examples:

A non-elementary

!

"

-polycycleAn elementary

!

"

-polycycle

p.14/4

O d


17/73

Open edges

An open edge of an

-polycycle is an edge on aboundary such that each of its end-vertices havedegree less than .

Examples

p.15/4

D iti th


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Decomposition theorem

Thm. Any

-polycycle is uniquely decomposed intoelementary

-polycycles along its bridges.

In other words, any

-polycycle is obtained by

gluing some elementary

-polycycles along openedges.

p.16/4

D iti th


19/73

Decomposition theorem

Thm. Any

-polycycle is uniquely decomposed intoelementary

-polycycles along its bridges.

In other words, any

-polycycle is obtained by

gluing some elementary

-polycycles along openedges.

p.16/4

S


20/73

Summary

Elementary

-polycycles provide a decompositionof

-polycycles.

In order for this to be useful, we have to classify the

elementary

-polycycles.

For non-elliptic cases, there is no hope of reasonableclassification (there is a continuum of elementary ones):

p.17/4


21/73

III. Classification

of elementary -polycycles

p.18/4

With at least one gon


22/73

With at least one -gon

All elementary

"

!

"

-polycycles, containing a

-gon,are those eight ones:

bonjour bonjour bonjour bonjour

bonjour bonjour bonjour bonjour

p.19/4

Totally elementary polycycle


23/73

Totally elementary polycycle

Call an elementary

"

-polycycle totally elementary if,after removing any face adjacent to a hole, one obtainsa non-elementary

"

-polycycle.

Examples:

A totally elementarypolycycle A non-totally elementarypolycycle p.20/4

Classification of totally elementary


24/73


Any totally elementary

"

!

"

-polycycle is one of:three isolated

-gons,

"

!

:

all ten triples of

-gons,

"

!

:

p.21/4



25/73


the following doubly infinite

!

"

-polycycle, denotedby

i

p & &q

r

s :

the infinite series ofi

p & &q

r

7, 0u

:

p.21/4

Classification methodology


26/73

Classification methodology

If an elementary polycycle is not totally elementary,then it is obtained from another elementary one withone face less.

So, from the list of elementary

"

!

"

-polycycleswith v faces, one gets the list of elementary

"

!

"

-polycycles with v6

w

faces.

p.22/4

Full classification


27/73

Full classification

Any elementary

"

!

"

-polycycle is one of:eight such polycycles containing

-gons

totally elementary polycycles

sporadic polycycles with

tow w

proper faces

six

"

!

"

-polycycles, infinite in one direction:

V

x

X

y

Y

p.23/4

Full classification


28/73

Full classification

w

E

e

infinite series obtained by taking two endings

of the above infinite polycycles and concatenating them.

See below three examples in the infinite series

X

y

p.23/4

Subcase of -polycycles


29/73

Subcase of $ -polycycles

Sporadic elementary

!

"

-polycycles:

3

3

p.24/4



30/73

Subcase of $ polycycles

p.24/4



31/73

Subcase of $ polycycles

The infinite series of elementary

!

"

-polycyclesV V

:

W

d

The only elementary infinite

!

"

-polycycle arei

p & & q

r

s

andV

p.24/4



32/73

Subcase of polycycles


!

"

-polycyclesi

p & & q

r

5 ,

u

"

:

p.24/4


33/73

IV. Classification


p.25/4

The classification


34/73

The classification

Any elementary

"

-polycycle is one of the followingeight:

p.26/4


35/73

V. Classification


p.27/4

The technique


36/73

q

Take an elementary

"

!

-polycycle. If

is a vertexon the boundary, then we can consider all possibleways to make this vertex an interior vertex in an

elementary

"

!

-polycycle.From the list of elementary

"

!

-polycycles with v

interior vertices, one can obtain the list of elementary

"

!

-polycycles with

v

6

w

interior vertices.

p.28/4

The classification


37/73

The classification

Any elementary

"

!

"

-polycycle is one of:!

sporadic

"

!

-polycycles.

three following infinite

"

!

-polycycles:

V:

Bonjour, ici la terre

X

:

Bonjour, ici la terre

Y:

p.29/4

The classification


38/73

the following

!

-valent doubly infinite

"

!

-polycycle,called snub -antiprism:

the infinite series of snub 0-antiprisms, 0u

(two0-gonal holes):

six infinite series of

"

!

-polycycles with one hole(they are obtained by concatenating endings V,

X

, Y)

p.29/4

The classification


39/73

Infinite seriesV Y

of elementary

"

!

-polycycles:

Infinite series

X

Y

of elementary

"

!

-polycycles:

p.29/4



40/73

p y y

Sporadic elementary

"

!

-polycycles:

p.30/4



41/73

p.30/4



42/73


"

!

-polycyclesV V

:

The only elementary infinite

"

!

-polycycles are V

and snub -antiprism.


"

!

-polycycles

snub0

-antiprisms,0

u

:

p.30/4


43/73

VI. Application

to extremalpolycycles

p.31/4

Definition


44/73

Given a finite

&

-polycycle

%

, denote byv

C

1

%

the number of interior vertices

and

%

the number of faces in

.

Fix

. An

&

-polycycle with

%

E

is calledextremal if it has maximal v C 1

%

among all

&

-polycycles with

%

E

.

Problem to find

j

5

, the maximal number of vertices.Fact: For fixed &

%

E

extremal polycycle hasalso maximal v C 1

%

,q C 1

%

(interior faces) and

minimalv

,

r

,

%

q &

0

E

v

l 9

For

&

=

"

"

,

"

,

"

, the question is trivial.m

authors, 1997: found

n

j

W

for o

w

(unique, partial

subgraph of Dodecahedron). p.32/4

Use of elementary polycycles


45/73

If a

&

-polycycle%

is decomposed into elementary

&

-polycycles

%

C

C

I

appearing C times, then onehas:

v

C

1

%

E

C

I

C

v

C

1

%

C

%

E

C

I

C

%

C

If one solves the Linear Programming problem

maximize CI C v C 1

%

C

with E CI C

%

C

and

C

and if

C

C

I

realizing the maximum can be realized as

&

-polycycle, then

j

5

can be found. p.33/4

Small extremal $ -polycycles


46/73

n

j

W

extremal components

1 0`

2 0`

`

3 1

4 2

5 3

W

p.34/4



47/73

n

j

W

extremal components

6 53

n

7 6i

W

8 83

d

9 103

W

10 123

p.34/4



48/73

n

j

W

extremal components

11 153

12 10

i

`

`

`

`

p.34/4

Extremal $ -polycycles


49/73

Thm: For any

u

w

, one has

n

j

W

E

m

F

z

w

D

w

#

F

z

w

D

w

"

!

F

z

w

Extremal polycycle realizing the extremum:

If

F

z

w

(unique):

If

F

z

w

(unique):

p.35/4



50/73

Extremal polycycle realizing the extremum:If

m

F

z

w

(unique):

If

F

z

w

(non-unique):

If #

F

z

w

(non-unique):

Otherwise (non-unique):

1

p.35/4



51/73

Theorem

If

w

F

z

w m

, then

W

j

n

E

{

9

W

|

except that

W

j

n

w m

E

m

and

W

j

n

w

E

.

Otherwise,

W

j

n

E

{

9

}

W

|

except that

W

j

n

E

{

9

W

|

for Ew

w

w "

w

m

"

" w

" "

" !

and

W

j

n

E

{

9

d

W

|

for

E

w !

w #

w

"

.

p.36/4

Non-elliptic case


52/73

For parabolic

&

-polycycles (i.e.

&

=

,

#

"

or

"

#

) the method of elementary polycycles fails (sincethere is no classification) but extremal animals of

Harary-Hazborth 1976 (proper ones, growing as aspiral) are extremal:

Hyperbolic cases are very difficult.

p.37/4


53/73

VII. Application

to non-extendiblepolycycles

p.38/4

Definition


54/73

A

&

-polycycle is called non-extendible if it is noproper subgraph of another

&

-polycycle. Examples:

Extendible

"

-polycycle

Non-extendible

"

"

-polycycle

p.39/4

Classification


55/73

Thm.: All non-extendible

&

-polycycles are:

Regular partitions

&

5

Four following examples:

"

-polycycle

"

!

-polycycle

"

-polycycle

"

-polycycle

For any

&

E

"

"

,

"

,

"

a continuum of infiniteones.

p.40/4


56/73

Finite non-extendible polycycles


57/73

Main lemma: all finite non-extendible

&

-polycyclesare elliptic, i.e.

5

6

So, we can use decomposition of non-extendible

&

-polycycles into elementary

&

-polycycles andthe classification of them.

Given an

&

-polycycle%

, the graph of its elementary

components is denoted byq

r

%

; its vertices are itselementary

&

-polycycles with two elementary

&

-polycycles adjacent if they share an edge:

E 1 B 2

p.42/4

Finite non-extendible polycycles


58/73

A finite

&

-polycycle%

is a

&

-polycycle if andonly ifq

r

%

is a tree.

Every tree is either an isolated vertex, or contains at

least one vertex of degreew

.One checks on this vertex that there is only twopossibilities:

p.42/4


59/73

VIII. Application

toface-regular spheres

p.43/4

Euler formula


60/73

Take a"

-valent plane map and denote by

the numberof faces having

edges.

Then one has the equality

w

E

s

W

#

D

So, every"

-valent plane map has at least one face ofsize less than

#

.

So,"

-valent plane graphs with faces of gonality at most!

have at mostw

faces,

have at most

vertices.

p.44/4

Face regular maps


61/73

A

-sphere is a"

-valent plane graphs, whose facesare - or -gonal.

Take

a

-sphere. Then:

is called

C

if every

-gonal face is adjacent toexactly

-gonal faces.

is called

if every -gonal face is adjacent to

exactly

-gonal faces.The subject of enumerating them is very large. Weintend to show non-trivial results obtained by using

decomposition into elementary polycycles.

Q

!

. So, if one removes all -gonal faces and all edgesbetween any two -gonal faces, then the result is a

"

-polycycle. p.45/4

Polycycles of $ -sphere


62/73

The set of!

-gonal faces of

!

-sphere

isdecomposed into elementary

!

"

-polycycles.

Let us see in the classification the elementary

polycycles that could be okThey should be finite (this eliminate

i

p & &q

r

s and V)

They should have some vertices of degree

(this

eliminates Dodecahedron and

i

p & &q

r

)It should be possible to fill open edges so as to haveno pending vertices of degree

.

p.46/4



63/73

NO

NO

NO

NO NO p.46/4



64/73

YES NO NO

YES

NO

p.46/4



65/73


!

"

-polycyclesV V

:

YES NO

NO

p.46/4

$ -sphere


66/73

The set of!

-gonal faces of

!

-sphere

isdecomposed into the following elementary

!

"

-polycycles:

W

The major skeleton p

of a

!

-sphere

is a"

-valent map, whose vertex-set consists of polycycles

and

W .

It consists ofqr

with the vertices

(of degree

)being removed.

p.47/4

$ -sphere


67/73

A

!

w

-sphere

w

p.47/4

$ -sphere


68/73

The decomposition into elementarypolycycles. p.47/4

$ -sphere


69/73

E1

C 1

C 3E 1

C 3

C 1

C 1

E 1

E 1

E 1

E 1

Their names in the classification of

!

"

-polycycles. p.47/4

$ -sphere


70/73

E1

C 3E 1

E 1

E 1

E 1

E 1

C 1

C 1

C 1

The graph

q

r

p.47/4

$ -sphere


71/73

E

1

C 3E 1

E 1

E 1

E 1

E 1

p

: eliminate

, so as to get a

"

-valentmap p.47/4

Results


72/73

For a

!

-sphere

, the gonality of faces of the"

-valentmap p

is at most

5

.

Proof: Take a -gonal face

. Denote by , and

the number of

!

"

-polycycles

,

W

and

incident to

.

Counting edges, one gets:

E

6

"

6

!

which implies u

6

.

But

6

is the gonality of the face correspondingto

in p

.

p.48/4

Results


73/73

For

o

w

, we have a finite number of

!

-spheres

.Proof: Take such a plane graph

.

The associated map p

is"

-valent with faces of

gonality at most!

.

So, the number of

!

"

-polycycles

and

W is atmost

.

The number of polycycles

is bounded as well.

This implies that the number of vertices of

is boundedand so, we have a finite number of spheres.

p.48/4

Documents

Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles