Mathias Dutour / UAB Dev. team UNICOS regular meeting 29 January 2009.
Mathieu Dutour and Michel Deza- Zigzags in plane graphs and generalizations
Transcript of 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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