Mathieu Dutour, Michel Deza and Mikhail Shtogrin- Polycycles and their boundaries

8/3/2019 Mathieu Dutour, Michel Deza and Mikhail Shtogrin- Polycycles and their boundaries

1/67

Polycycles and

their boundariesMathieu Dutour

ENS/CNRS, Paris and Hebrew University, Jerusalem

Michel Deza

ENS/CNRS, Paris and ISM, Tokyo

and Mikhail Shtogrin

Steklov Institute, Moscow

p.1/4


2/67

I. -polycycles

p.2/4


3/67

Polycycles

A finite

-polycycle is a plane

-connected finite graph,such that :(i) all interior faces are (combinatorial) -gons,

(ii) all interior vertices are of degree

,(iii) all boundary vertices are of degree

or

.

p.3/4


4/67

Theorem

The skeleton of a plane graph is the graph formed by itsvertices and edges.Theorem

A

-polycycle is determined by its skeleton with the exception of thePlatonic solids, for which any of their faces can play role of exterior one

an unauthorized plane embedding

p.4/4


5/67

and -polycycles

(i) Any

-polycycleis one of the following

cases:

(ii) Any

-polycyclebelongs to the following

cases:

or belong to the following infinite family of

-polycycles:

So, those two cases are trivial.

p.5/4


6/67

Boundary sequences

The boundary sequence is the sequence of degrees(2 or 3) of the vertices of the boundary.

Associated sequence is3323223233232223

The boundary sequence is defined only up to action of

, i.e. the dihedral group of order

generated bycyclic shift and reflexion.

The invariant given by the boundary sequence is thesmallest (by the lexicographic order) representative ofthe all possible boundary sequences.

p.6/4


7/67

Enumeration of -polycycles

There exist a large litterature on enumeration of

-polycycles; they are called benzenoids.

benzene

naphtalene

azulene

Algorithm for enumerating

-polycycles with -gons:

1. Compute the list of all

-gonal patches with

-gons

2. Add a -gon to it in all possible ways

3. Compute invariants like their smallest (by thelexicographic order) boundary sequence

4. Keep a list of nonisomorph representatives (we usehere the program nauty by Brendan Mc Kay)

p.7/4


8/67

Enumeration of small -polycycles

1 1 6 18 11 1337

2 1 7 35 12 3524

3 2 8 87 13 92624 4 9 206 14 24772

5 7 10 527 15 66402

p.8/4


9/67

Benzenoids of lattice type

We say that a

-polycyle has lattice type if its skeleton isa partial subgraph of the skeleton of the partition of theplane into hexagons.

Such

-polycycles are uniquely defined by theirboundary sequence.M. Deza, P.W. Fowler, V.P. Grishukhin, Allowed boundary sequences for

fused polycyclic patches, and related algorithmic problems, Journal of

Chemical Information and Computer science 41-2 (2001) 300308.

p.9/4


10/67

II. -polycycles

with given boundary

p.10/4


11/67

The filling problem

Does there exist

-polycycles with given boundarysequence?

If yes, is this

-polycycle unique?

Find an algorithm for solving those problemscomputationally.

Remind, that the cases

or

are trivial.Let

; consider, for example, the sequence

p.11/4


12/67

The filling problem

Does there exist

-polycycles with given boundarysequence?

If yes, is this

-polycycle unique?

Find an algorithm for solving those problemscomputationally.

Remind, that the cases

or

are trivial.Let

; consider, for example, the sequence

p.11/4


13/67

The case of -polycycles

Two

-polycycles with the same boundary.

Boundary sequence:

,

vertices of degree

,

, resp.Symmetry groups: of boundary:

, of polycycles:

.Fillings:

pentagons,

interior points.

p.12/4


14/67

The case of -polycycles

Two

-polycycles with the same boundary.

Boundary sequence:

,

vertices of degree

,

, resp.Symmetry groups: of boundary:

, of polycycles:

.Fillings:

hexagons,

interior points.

p.13/4

N i f


15/67

Non-uniqueness for any

Boundary sequence is:

;

vertices of degree

and

of degree

.

Symmetry groups are:of boundary:

,of polycycles:

.

This domain is filled in two ways (by

-gons;

interior-valent vertices).Thm.: The boundary does not determine

-polycycle if

. Conj.: but it determines it if the filling is by less than

-gons.

p.14/4

E l f l f l l


16/67

Euler formula for -polycycles

Let

be a

-polycycle. Let , be the number ofvertices of degree

or

on the boundary. Let

the number

of -gonal faces and the number of interior vertices

Theorem(i) one has the relations

(ii) If

, then

and

are determined by the boundary sequence.

(iii) If

, then

.

p.15/4

P ibl filli


17/67

Possible filling

Let us illustrate the algorithm for the simplest case

.In some cases we can complete the patch directly.

p.16/4

P ibl filli


18/67

Possible filling



p.16/4

P ibl filli


19/67

Possible filling



But in some cases more is needed:

p.16/4

Diff t ibl ti


20/67

Different possible options

p.17/4



21/67


or

p.17/4



22/67


or

p.17/4

Algorithm


23/67

Algorithm

A patch of -gonal faces is a group of faces with one ormore boundaries.Take a boundary of a patch of faces. Then:

1. Take a pair of vertices of degree

on the boundary andconsider all possible completions to form a -gon.

2. Every possible case define another patch of faces.

Depending on the choice, the patch will have one ormore boundaries.

3. For any of those boundaries, reapply the algorithm.

This algorithm is a tree search, since we consider allpossible cases.

p.18/4

An example of a search


24/67


p.19/4



25/67


p.19/4



26/67


p.19/4



27/67


p.19/4



28/67


p.19/4



29/67


p.19/4



30/67


p.19/4

Another possible search


31/67


p.20/4



32/67


p.20/4



33/67


p.20/4



34/67


p.20/4



35/67

p a

p.20/4



36/67

p

p.20/4

Possible speedups


37/67

p p

Limitation of tree size:

Do all automatic fillings when there are some.

Then, we can select the pair of consecutive vertices

of degree

with maximal distance between them.

Kill some branches if :

or are not non-negative integers (they are

computed from the boundary sequence by Eulerformula).

two consecutive vertices of degree

do not admit

any extension by a

-gon.

The combination of those tricks is insufficient in manycases. For the enumeration of the maps

below, this

is the critical bottleneck. p.21/4


38/67

III. maps of -gons

with a ring of -gons

p.22/4

The problem


39/67

p

A

denotes a

-valent plane graph having only-gonal and -gonal faces, such that the -gonal faces form

a ring, i.e. a simple cycle, of length .

Theorem: One has the equation

with

and

being the number of interior vertices in two

-polycycles defines by the ring of

-gons.

M. Deza and V.P. Grishukhin, Maps of -gons with a ring of -gons,

Bull. of Institute of Combinatorics and its Applications 34 (2002) 99110.

p.23/4

Classification theorem


40/67

Main Theorem

Besides the cases

=

and

with

, all such maps

are known;

If

, then the map is

; from now, let

.If

, two possibilities:

p.24/4

Case


41/67

If

, two possibilities:

and an infinite serie

p.25/4

Case


42/67

If

, then this is Dodecahedron

If

, then five possibilities:

5,

;6,66,

;6,6 6,

;6,6

8,

;6,6 10,

;6,6

If

, then ten possibilities

If

, we expect infinity of possibilities p.26/4

All


43/67

4,

;8,8 10,

;12,10 12,

;10,14

12,

;13,11 12,

;12,12 12,

;12,12 p.27/4


44/67

16,

;14,14 16,

;14,14

20,

;16,16 20,

;16,16 p.28/4

Case


45/67

If

, then

. There are four possibilities:

12,

;4,4 12,

;6,6

12,

;7,7 12,

;6,6 p.29/4

Two remaining undecided cases


46/67

If

, then

and

. Two examples:

28,

;8,8 30,

;9,9

The remaining undecided case is

with

.

Hadjuk and Sotk found an infinity of maps

,

Madaras and Sotk found infinity of maps

for

and

,

. p.30/4

Enumeration techniques


47/67

Harmuth enumerated all

-valent plane graphs with atmost

vertices, faces of gonality

or

and such thatevery faces of gonality

is adjacent to two faces ofgonality

(i.e.

-gons are organised into disjoint simple

cycles). It gives all

with

.

Remaining case

is treated by followingalgorithm:

Generating

patches

Adding ring of

gons

completing (if

possible). p.31/4

Known


48/67

3,

;9,9 4,

;10,10 8,

;10,18

9,

;19,11 10,

;10,2210,

;14,18

p.32/4


49/67

11,

;20,14 12,

;26,10 12,

;14,22

12,

;14,22 12,

;21,15 13,

;15,23 p.33/4


50/67

14,

;28,12 16,

;30,14

18,

;14,34 18,

;33,15 p.34/4

Known


51/67

6,

;10,20 10,

;34,8

12,

;40,8 12,

;39,9 p.35/4


52/67

12,

;38,10 12,

;38,10

12,

;38,10 p.36/4

Known


53/67

2,

;10,10 6,

;12,24 6,

;14,22

6,

;13,23 6,

;14,22 6,

;12,24 p.37/4


54/67

6,

;15,21 12,

;11,4914,

;58,10

14,

;11,57 14,

;11,57 14,

;11,57 p.38/4

All parameters


55/67

maps

(full.)

maps

(Dode.)

(full.)

(azu.)

?

(azu.)

p.39/4


56/67

IV. Generalizations

p.40/4

Several rings


57/67

A

denotes a

-valent plane graph with

-gonsand -gons, where -gons form

rings of length , . . . ,

(equiv. each -gon is adjacent exactly to two -gons).

Theorem: One has the equation

, where

is the number of vertices incident to

-gonal faces and

the number of vertices incident to

-gonal faces.

finiteness for

,

,

but we have infinity for

and, possibly, for

,

.

p.41/4

The case = (fullerenes)


58/67

All maps

are:

five maps with one ring of

-gons,

following three maps with two rings of

-gons:

; 32

; 38

; 40

p.42/4

Two rings of -gons filled by -gons


59/67

; 44 ; 44

; 60

; 60

; 68

; 68 p.43/4

Two rings of -gons filled by -gons


60/67

; 68

; 68

; 68

; 68

; 76

; 68

p.44/4

Remaining graphs

(azulenoids)


61/67

; 76 ; 80

; 92

; 100 p.45/4

The case = (fullerenes)


62/67

All maps

are:four maps with exactly one ring of

-gons,

the maps:

special map

infinite family:

triples of

pentagons

infinite family:

concentric

-rings of

hexagons p.46/4

Infinite families


63/67

For any

, there exists a map

(with

-ringsof

-gons) and a map

(with

-rings of

-gons)

p.47/4

Infinite families


64/67

For any


(with

-ringsof

-gons) and a map

(with

-rings of

-gons)

p.47/4

Infinite families


65/67

For any


(with

-ringsof

-gons) and a map

(with

-rings of

-gons)

p.47/4

Infinite families


66/67

For any


(with

-ringsof

-gons) and a map

(with

-rings of

-gons)

p.47/4

-valent maps


67/67

A

denotes a

-valent map with

-gons and

-gonsonly, where -gons form a ring of length .

The only

is -gonal antiprism.

All

are:

; 10

; 12 ; 12

; 14

There is only one other

; it has two rings of

-gons,

vertices and symmetry

. p.48/4

Documents

Mathieu Dutour, Michel Deza and Mikhail Shtogrin- Polycycles and their boundaries