mathieu dutour and michel deza- zigzags in plane graphs and generalizations

8/3/2019 Mathieu Dutour and Michel Deza- Zigzags in plane graphs and generalizations

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I. Simple

two-faced

polyhedra

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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 byits 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 is planar and

-connected.(ii)

and

are in the same combinatorial type if and only if

is isomorphic to

.

The dual graph

of a plane graph

is the plane graphformed by the faces of

, with two faces adjacent if theyshare an edge.

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Simple two-faced polyhedra

A polyhedron is called simple if all its vertices are

-valent.If one denote the number of faces of gonality

, thenEulers relation take the form:

A simple planar graph is called two-faced if the gonality ofits faces has only two possible values:

and

, where

.

We consider mainly classes , i.e. simple planar graphswith vertices and

;there are

cases:

, , .

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-connectedness

Theorem

(i) Any 3-valent plane graph without (>6)-gonal faces is

-connected.(ii) Moreover, any 3-valent plane graph without (>6)-gonal

faces is

-connected except of the following serie

:

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-connectedness

Theorem

(i) Any 3-valent plane graph without (>6)-gonal faces is

-connected.(ii) Moreover, any 3-valent plane graph without (>6)-gonal

faces is

-connected except of the following serie

:

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Zigzags

Take two edges

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Zigzags

Continue it leftright alternatively ....

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Zigzags

A selfintersecting zigzag

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Zigzags

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

z=10 , 162zvector 2,0 p.9/4


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Intersection Types

Let

and

be (possibly,

) zigzags of a plane graph

and let an orientation be selected on them. An edge ofintersection

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.10/4


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D li d


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Duality and types

TheoremThe zigzags of a plane graph

are in one-to-one correspondence with zigzags of

. The length is

preserved, but intersection of type I and II are interchanged.TheoremLet

be a plane graph; for any orientation of all zigzags of

, we have: (i) The number of edges of type II, which are incident to any xed vertex , is even.(ii) The number of edges of type I, which are incident to any xed face , is even.

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Bi i h


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Bipartite graphs

Remark A plane graph is bipartite if and only if its faces have even gonality.Theorem (Shank-Shtogrin)

For any planar bipartite graph

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

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Zi ti f h


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Zigzag properties of a graph

q -uniform : all zigzags have the same length and signature,

q -transitive : symmetry group is transitive on zigzags,

q -knotted : there is only one zigzag,

q -balanced : all zigzags of the same length and signature, haveidentical intersection vectors.

All known

-uniform

-valent graphs are

-balanced.

(

-vertices

,

"

(

1

) $

!

&

(

),

"

) $

&

'

(

1

! (

),

"

'

#

Smallest (among all )

-valent, all &

, all

) -unbalanced )

-valent graphs

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Perfect matching on graphs


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Perfect matching on graphs

Let

be a -knotted graph

.

(i)

with

. If

then the edges of type I forma perfect matching

(iii) every face incident to two orzero edges of

(iv) two faces,

and

are in-cident to zero edges of

,

is organized around themin concentric circles.

F2

F1

M. Deza, M. Dutour and P.W. Fowler, Zigzags, Railroads and Knots in Fullerenes , (2002).

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Railroads


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Railroads

A railroad in graph ,

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

0

&

%

0

$

#

Railroads, as well as zigzags, can be self-intersecting(doubly or triply). A graph is called tight if it has no railroad.

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with triply self int railroad


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with triply self-int. railroad

It is smallest such . Green railroad also triply self-int. p.21/4


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Triply intersecting railroad in


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Triply intersecting railroad in

Conjecture: a railroad-curve of any appears in some .

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IPR


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Comparing graphs


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p g g p

q

max # of zigzags in tight

all tight with simple zigzags all tight Cube, Tr. Oct. examples(?)

int. size of simple zigzags any even ,

,

any even

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Parametrizing graphs


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g g p

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

Goldberg (1937) All

, or of symmetry ( , ), ( ,) or (

,

) are given by Goldberg-Coxeterconstruction

.

Fowler and al. (1988) All of symmetry

2 ,

' orare described in terms of parameters.Graver (1999) All can be encoded by

integerparameters.

Thurston (1998) The are parametrized by

complexparameters.

Sah (1994) Thurstons result implies that the Nrs of

, ,

,

%

,

. p.32/4

Goldberg-Coxeter construction


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g

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 thecycles structure of

.

For tight of symmetry

or

this gives

,

or

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

-valent plane graphs , submitted

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The structure of graphs


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triangles in

The corresponding trian-gulation

The graph

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Conjecture on , parameters


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For tight graphs

of symmetry

%

,

%

or

%

the-vector is of the form

with

or

with

A knotted of such symmetry has symmetry

% .

if there is a knotted of symmetry

% , then

is the

product of at most

primes

First

-knotted

of sym-metry

.

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Lins trialities


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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.41/4

Example: Tetrahedron


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two Lins maps on projective plane.

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VI. Zigzags

on -dimensionalcomplexes

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Zigzag of reg. and semireg. -polytopes


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-polytope -vector

Dodecahedron

-cell

-cell

-simplex=

-cross-polytope=

octicosahedric polytope

snub

-cell

=Med( )

=Half-

-Cube

=Schli polytope (in

)

=Gosset polytope (in

)

(

roots of

)

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mathieu dutour and michel deza- zigzags in plane graphs and generalizations

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