Michel Deza, Mikhail Shtogrin, Mathieu Dutour and Patrick Fowler- Zigzags and Central Circuits for 3- or 4-valent plane graphs and generalizations

8/3/2019 Michel Deza, Mikhail Shtogrin, Mathieu Dutour and Patrick Fowler- Zigzags and Central Circuits for 3- or 4-valent plane graphs and generalizations

1/83

Zigzags and Central Circuits

for - or -valent plane graphs

and generalizationsMichel Deza

ENS/CNRS, Paris and ISM, Tokyo

Mathieu Dutour

Hebrew University, Jerusalem

Mikhail Shtogrin

Steklov Institute, Moscow

and Patrick Fowler

Exeter University

p.1/5


2/83

I. -valent

two-faced polyhedra

p.2/5


3/83

Polyhedra and planar graphs

A graph is called

-connected if after removing any set of

vertices it remains connected.

The skeleton of a polytope

is the graph

formed by

its vertices, with two vertices adjacent if they generate aface of

.Theorem (Steinitz)

(i) A graph

is the skeleton of a

-polytope if and only if it isplanar and

-connected.(ii)

and

are in the samecombinatorial typeif and only if

is isomorphic to

.

A planar graph is represented as Schlegel diagram, the pro-

gram used for this is CaGe by G. Brinkmann, O. Delgado, A.

Dress and T. Harmuth. p.3/5


4/83

-valent two-faced polyhedra

The Euler formula for plane graphs

,take the following form for

-valent graphs:

if

and

if

With

the number of faces of gonality

.A

-valent plane graph is called two-faced if the gonality ofits faces has only two possible values and

.

-valent plane graphs with

vertices and faces ofgonality and

(classes ),

-valent plane graphs with vertices and faces of

gonality

or

(octahedrites)

p.4/5


5/83

Classes and their generation

Polyhedra Exist if and only if

"

#

$

%

&

' '

(

'

'

"

#

$

)

&

0

(

1

1

(fullerenes)

"

#

$

2

&

# 3

(

'

'

octahedrite )

"

#

$

%

&

0

(

)

Generation programs

1.

-valent: CPF for two-faced maps on the sphere by T.Harmuth

-valent: CGF for two-faced maps on surfaces of genus4

by T. Harmuth

2.

-valent: ENU by T. Heidemeier

3. General: plantri by G. Brinkmann and B. McKay

p.5/5


6/83

Point groups

(point group)

(combinatorial group)Theorem (Mani, 1971)Given a

-connected planar graph

, there exist a

-polytope

, whose group of isometries is isomorphic to and

.

q For octahedrites: (

,

,

), (

,

,

,

! ), (#

,#

$

,#

), (#

%

,#

%

$

,#

%

), (#

!

,#

!

$

,#

!

), ((

,(

). (Dezaand al.)

q For

: (#

,#

,#

$), ()

,)

$) (Fowler and al.)

q For

: (

,

,

), (

,

,

), (#

,#

$

,#

), (#

%

,#

%

$,#

%

), (#

4 ,#

4

), ((

,(

) (Deza and al.)

q For6

: (

,

,

), (

,

,

,

! ), (

% ,

%

,

%

,

4 ),

(#

,#

,#

$

), (#

%

,#

%

,#

%

$

), (#

7

,#

7

,#

7

$

), (#

4

,#

4

,#

4

$), ()

,)

$,)

), (

,

) (Fowler and al.) p.6/5


7/83

-connectedness

Theorem

(i) Any octahedrite is

-connected.

(ii) Any

-valent plane graph without (

)-gonal faces is

-connected.

(iii) Moreover, any


)-gonalfaces is

-connected except of the following serie

:

p.7/5


8/83

-connectedness

Theorem


-connected.

(ii) Any


)-gonal faces is

-connected.

(iii) Moreover, any


)-gonalfaces is


:

p.7/5


9/83

-connectedness

Theorem


-connected.

(ii) Any


)-gonal faces is

-connected.

(iii) Moreover, any


)-gonal

faces is


:

p.7/5


10/83

Medial Graph

Given a plane graph

, the

-valent plane graph

isdefined as the graph having as vertices the edges of

withtwo vertices adjacent if and only if they share a vertex and

belong to a common face.

p.8/5


11/83

Medial Graph

Given a plane graph

, the

-valent plane graph

isdefined as the graph having as vertices the edges of

withtwo vertices adjacent if and only if they share a vertex and

belong to a common face.

p.8/5


12/83

Inverse medial graph

If

is a

-valent plane graph, then there exist exactly twographs

and

such that

.

p.9/5


13/83


If

is a


and

such that

.

p.9/5


14/83


If

is a


and

such that

.

p.9/5


15/83

II. Zigzags

and

Central circuits

p.10/5


16/83

Central circuits

A 4valent plane graph G

p.11/5


17/83

Central circuits

Take an edge of G

p.11/5

C l i i


18/83

Central circuits

Continue it straight ahead ...

p.11/5

C t l i it


19/83

Central circuits

... until the end

p.11/5

C t l i it


20/83

Central circuits

A selfintersecting central circuit

p.11/5

C t l i it


21/83

Central circuits

A partition of edges of G

CC=4 , 6, 82 p.11/5

Zigzags


22/83

Zigzags

A plane graph G

p.12/5

Zigzags


23/83

Zigzags

Take two edges

p.12/5

Zigzags


24/83

Zigzags

Continue it leftright alternatively ....

p.12/5

Zigzags


25/83

Zigzags

... until we come back

p.12/5

Zigzags


26/83

Zigzags

A selfintersecting zigzag

p.12/5

Zigzags


27/83

Zigzags

A double covering of 18 edges: 10+10+16

z=10 , 162zvector 2,0 p.12/5

Intersection Types for zigzags


28/83


Let

and

be (possibly,

) zigzags of a plane graphand let an orientation be selected on them. An edge of

intersection

is called of type I or type II, if

and

traverse in opposite or same direction, respectively

ee

Type I Type II

ZZZ Z

p.13/5



29/83


Let

and

be (possibly,

) zigzags of a plane graphand let an orientation be selected on them. An edge of

intersection

is called of type I or type II, if

and

traverse in opposite or same direction, respectively

ee

Type I Type II

ZZZ Z

The types of self-intersection depends on orientation

chosen on zigzags except if

:

Type IIType I

p.13/5

Intersection Types for central circuits


30/83


Let

be a

-valent plane graph

p.14/5



31/83


Take

( ),

( ) a bipartition of the face-set of

p.14/5



32/83


Let

and

be two central circuits of

and let anorientation be selected on them.

C

C

p.14/5



33/83


Local View on a vertex

and type

Type IIType I

v

C C C

v

C

p.14/5



34/83


and

have

intersections I and

intersection II

I II

II

I

p.14/5

Medial, zigzags and central circuits


35/83

a , g ag a a

Zigzags of a plane graph

are in one-to-onecorrespondence with zigzags of

.

Types are interchanged

p.15/5

Medial, zigzags and central circuits


36/83

, g g

Zigzags of a plane graph

are in one to onecorrespondence with central circuits of

.

p.15/5

Intersection two simple ZC-circuits


37/83

p

For the class of graph

the size of the intersection oftwo simple zigzags belongs to

.

For classes of octahedrites, graph

or graph6

the

size of the intersection of two simple ZC-circuits can beany even number.

Two simple zigzags

of a graph6

with

.

On surfaces ofgenus 4

, theintersection can be

odd. p.16/5

Bipartite graphs


38/83

p g p

Remark A plane graph isbipartite if and only if its faces haveeven gonality.Theorem (Shank-Shtogrin)For any planar bipartite graph

there exist an orientation ofzigzags, with respect to which each edge has type I.

p.17/5

Perfect matching on graphs


39/83

g g p

Let

be a graph

with one

zigzag with self-intersection numbers

.

(i)

. If

then

the edges of self-intersection oftype I form a perfect matching

(ii) every face incident to

or

edges of

(iii) two faces,

and

are free of

,

is organized aroundthem in concentric circles.

F2

F1

M. Deza, M. Dutour and P.W. Fowler, Zigzags, Railroads and Knots in

Fullerenes, Journal of Chemical Information and Computer Sciences, in

press.

p.18/5


40/83

III. railroad

structureand tightness

p.19/5

Railroads, -valent case


41/83

A railroad in an octahedrite is a circuit of square faces, suchthat any of them is adjacent to its neighbors on oppositefaces. Any railroad is bordered by two central circuits

4

#

Railroads, as well as central circuits, can be

self-intersecting. A graph is called tight if it has no railroad. p.20/5

Railroads, -valent case


42/83

A railroad in graph

,

6

is a circuit of hexagonalfaces, such that any of them is adjacent to its neighbors onopposite faces. Any railroad is bordered by two zigzags.

!

#

%

!

Railroads, as well as zigzags, can be self-intersecting

(doubly or triply). A graph is called tight if it has no railroad. p.21/5

Triple self-intersection


43/83

4 4

#

%

It is smallest such

.

6

4

%

Conjecture: It is smallestsuch

6

p.22/5

Removing central circuits


44/83

Take a

-valent plane graph

and a central circuit

p.23/5



45/83

Remove the edges of the central circuit

p.23/5



46/83

Remove the vertices of degree

or

p.23/5

Removing zigzags


47/83

Take a plane graph

and a zigzag

p.24/5

Removing zigzags


48/83

Go to the medial

p.24/5

Removing zigzags


49/83

Remove the central circuit

p.24/5

Removing zigzags


50/83

Take one (out of two) inverse medial graph

p.24/5

Extremal problem


51/83

Given a class of tight graphs (octahedrites, graphs

),there exist a constant

such that any element of the classhas at most

ZC-circuits.

Every tight octahedrite has at most

central circuits.Proof method: Local analysis + case by case analysis.

Every tight

has exactly

zigzags.Proof method: uses an algebraic formalism on thegraphs

.

Every tight

has at most

zigzags.

Conjecture: The correct upper bound is

. checked for

Every tight6

has at most 6

zigzags.

Attempted proof: uses a local analysis on zigzags. p.25/5

Tight with simple central circuits


52/83

Theorem 1 There is exactly

tight octahedrites with simplecentral circuits.

Proof method: after removing a central circuit, the obtained

graph has faces of gonality at most

.

6(

12(

'

12

#

'

14#

'

p.26/5

Tight with simple central circuits


53/83

Theorem 1 There is exactly

tight octahedrites with simplecentral circuits.

Proof method: after removing a central circuit, the obtained

graph has faces of gonality at most

.

20

#

1

22#

30

(

32

#

'

'

p.26/5

Tight with simple zigzags


54/83

All tight

have simple zigzagsInfinity of such graphs

There are exactly

tight graph

with simple zigzags:

Cube and Truncated Octahedron=

.Proof method: the sizeof intersection of two sim-ple zigzags is at most

. There is at most

zigzags.Upper bound on .

6(

,

!

24(

,

4

There is at least

tight graphs6

with simple zigzags.G. Brinkmann and T. Harmuth computation of fullereneswith simple zigzags up to

vertices.

p.27/5



55/83

,

4

)

$,

#

% ,

#

%

,

,

#

$

,

!

p.28/5



56/83

)

,

)

,

4

4

,

7

p.28/5


57/83

IV. Goldberg-Coxeter

construction

p.29/5

The construction


58/83

Take a

- or

-valent plane graph

. The graph

isformed of triangles or squares.

Break the triangles or squares into pieces according to

parameter

.

3valent case

k=5

l=2

l=2

k=5

4valent case p.30/5

Gluing the pieces


59/83

Glue the pieces together in a coherent way.We obtain another triangulation or quadrangulation ofthe plane.

p.31/5

Final steps


60/83

Go to the dual and obtain a

- or

-valent plane graph,which is denoted

and called

Goldberg-Coxeter construction.

The construction works for any

- or

-valent map onoriented surface.

Operation

on Tetrahedron, Cube and Dodecahedron

p.32/5

Goldberg-Coxeter for Cube1 0 1 1


61/83

1,0 1,1

2,0 2,1

p.33/5

Goldberg-Coxeter for Octahedron

1 0 1 1


62/83

1,0 1,1

2,0 2,1

p.34/5

Properties


63/83

One associates

% (Eisenstein integer) or

(Gaussian integer) to the pair

in

- or

-valent case.

If one writes

instead of

, then one

has:

If

has

vertices, then

has

vertices if

is

-valent,

vertices if

is

-valent.

If

has a plane of symmetry, we reduce to

.

has all rotational symmetries of

and all

symmetries if

or

.

p.35/5

The special case


64/83

Any ZC-circuit of

corresponds to

ZC-circuits of

with length multiplied by

.

If the ZC-vector of

is

, then the ZC-vector

of

is

.

p.36/5

The special case


65/83

Any ZC-circuit of

corresponds to

ZC-circuits of


.

If the ZC-vector of

is


of

is

.

p.36/5

The special case


66/83

Any ZC-circuit of

corresponds to

ZC-circuits of


.

If the ZC-vector of

is


of

is

.

p.36/5

-product formalism


67/83

Given a

-valent plane graph

, the zigzags of theGoldberg-Coxeter construction of

are obtained by:

Associating to

two elements

and

of a groupcalled moving group,

Computing the value of the

-product

,

The lengths of zigzags are obtained by computing thecycle structure of

.

For

#

with 4

, this gives

,

or 6

zigzags.M. Dutour and M. Deza, Goldberg-Coxeter construction for

- or

-valent

plane graphs, Electronic Journal of Combinatorics, 11-1 (2004) R20.

p.37/5

Illustration


68/83

For any ZC-circuit of

, there exist

=

-valent case

=

-valent case

The [ZC]-vector of

is the vector

where is the number of ZC-circuits with order .

If 4

, then

has

zigzags if

and

otherwise.For Truncated Icosidodecahedron, possible [ZC]:

%

!

%

6

!

6

!

4

4

6

!

6

4

6

% 4

%

6

p.38/5


69/83

V. Parametrizing

graphs

p.39/5

Parametrizing graphs


70/83

Idea: the hexagons are of zero curvature, it suffices to giverelative positions of faces of non-zero curvature.

Goldberg (1937) All

,

or6

of symmetry ()

,)

$), ((

,(

) or (

,

) are given by Goldberg-Coxeterconstruction

.

Fowler and al. (1988) All6

of symmetry#

7 ,#

4 or)

are described in terms of

parameters.Graver (1999) All

6

can be encoded by

integerparameters.

Thurston (1998) The6

are parametrized by

complexparameters.

Sah (1994) Thurstons result implies that the Nrs of

,

,6

,

%

,

. p.40/5

The structure of graphs


71/83

triangles in

The corresponding trian-gulation

The graph

#

$

p.41/5

- and railroad-structure of graphs


72/83

All zigzags and railroads are simple.

The -vector is of the form

%

%

with

the number of railroads is

%

.

has

zigzags with equality if and only if it is tight.

If

is tight, then

%

(so, each zigzag is aHamiltonian circuit).

All

are tight if and only if

! is prime.

There exists a tight

if and only if

! is odd.

p.42/5

General theory


73/83

Extensions:

-valent or

-valent graphs.

Classes of graphs with fixed

,

.Classes with a fixed symetry.

Maps on surfaces.

Dictionnary

-valent graph

'

-valent graph

ring Eisenstein integers

Gaussian integers

Euler formula

&

#

'

&

0

zero-curvature hexagons squares

ZC-circuits zigzags central circuits

Operation leapfrog graph medial graph

p.43/5

Number of parameters


74/83

Octahedrites:

Group

'

%

)

#

Graphs

:Groups

#

Graphs'

:

Group

'

%

#

Graph

1

:

Group

# 3

%

'

'

%

2

#

If there is just one parameter, then this is Goldberg-Coxeterconstruction (of Octahedron, Tetrahedron, Cube,Dodecahedron for octahedrite,

,

,6

, respectively).

p.44/5

Conjecture on

#

%

#

%

#

%

$

#

4

#

4

(

(

are described by two


75/83

are described by two

complex parameters. They exists if and only if

and

.

with one zigzag The defining triangles

!

#

$

exists if and only if

%

&

'

!

(

)1

2

3 4

$

,% 5 (.

If % increases, then part of

$

amongst

!

!

$

goes to 100%

p.45/5

More conjectures

All

with only simple zigzags are:


76/83

,

and

the family of

#

%

with parameters

and

with

and

%

%

They have symmetry#

%

$ or(

or#

4

Any

#

%

with one zigzag is a

#

%

.

For tight graphs

#

%

the -vector is of the form

with

or

with

Tight

#

%

$

exist if and only if

, they arez-transitive with

4

% 4

iff

and, otherwise,

!

iff

p.46/5


77/83

VI. Zigzagson

surfaces

p.47/5

Klein and Dyck map


78/83

Klein map:

Dyck map:

4

Zigzags (and central circuits), being local notions, aredefined on any surface, even on non-orientable ones.

Goldberg-Coxeter and parameter constructions aredefined only on oriented surfaces.

p.48/5

Lins trialities


79/83

our notation notation in [1] notation in [2]

gem

dual gem

phial gem

skew-dual gem

skew gem

skew-phial gem

Jones, Thornton (1987): those are only good dualities.

1. S. Lins, Graph-Encoded Maps, J. CombinatorialTheory Ser. B 32 (1982) 171181.

2. K. Anderson and D.B. Surowski, Coxeter-Petrie Complexes

of Regular Maps, European J. of Combinatorics 23-8 (2002) 861880.

p.49/5

Example: Tetrahedron


80/83

)

)

two Lins maps on projective plane.

p.50/5

Bipartite skeleton case

8 7


81/83

4

1

6

8 7

5

2

3

3 6

3

8 7

5 6 4

1 2

3 5

4

8 7

6

Two representation of

: on Torus and as a Cube

with cyclic orientation of vertices (marked by ) reversed.

Theorem

For bipartite graph embedded in oriented surface, the skew operation is,

in fact, reversing orientation of one of the part of the bipartition.

p.51/5

Prisms and antiprisms


82/83

Let denotes the Euler characteristic.Conjecture

has 4

and is oriented

iff

is even;

has

4

and isnon-oriented.

has

4

and isnon-oriented;

has

4

and isoriented.

p.52/5

The End


83/83

Removing

central circuits of

!

.

p.53/5

Documents

Michel Deza, Mikhail Shtogrin, Mathieu Dutour and Patrick Fowler- Zigzags and Central Circuits for 3- or 4-valent plane graphs and generalizations