michel deza, mathieu dutour sikiric and mikhail shtogrin- elementary polycycles
TRANSCRIPT
-
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
2/73
I. -polycycles
p.2/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
3/73
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
4/73
Examples with one hole
A
!
"
-polycycle A
"
!
-polycycle
A
"
!
-polycycle A
"
#
-polycycle
p.4/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
5/73
Examples with two hole or more
A
"
-polycycle A
!
"
-polycycle
p.5/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
6/73
$ -polycycles
Any
"
"
-polycycle is one of the following
Tetrahedron (with no hole):
"
following polycycles (with one hole):
p.6/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
7/73
$ -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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
8/73
$ -polycycles
The infinite family%
&
( 0
1 (with two holes)
Following two infinite
"
-polycycles:
singly infinite polycycle doubly infinite polycycle
p.7/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
9/73
$ -polycycles
Octahedron (with no hole):
Following polycycles (with one hole)
p.8/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
10/73
$ -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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
15/73
II. Decomposition
into elementary
polycycles
p.13/4
El l l
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
18/73
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
D iti th
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
21/73
III. Classification
of elementary -polycycles
p.18/4
With at least one gon
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
24/73
Classification of totally elementary
Any totally elementary
"
!
"
-polycycle is one of:three isolated
-gons,
"
!
:
all ten triples of
-gons,
"
!
:
p.21/4
Classification of totally elementary
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
25/73
Classification of totally elementary
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
29/73
Subcase of $ -polycycles
Sporadic elementary
!
"
-polycycles:
3
3
p.24/4
Subcase of -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
30/73
Subcase of $ polycycles
p.24/4
Subcase of -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
Subcase of $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
32/73
Subcase of polycycles
The infinite series of elementary
!
"
-polycyclesi
p & & q
r
5 ,
u
"
:
p.24/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
33/73
IV. Classification
of elementary -polycycles
p.25/4
The classification
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
34/73
The classification
Any elementary
"
-polycycle is one of the followingeight:
p.26/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
35/73
V. Classification
of elementary -polycycles
p.27/4
The technique
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
39/73
Infinite seriesV Y
of elementary
"
!
-polycycles:
Infinite series
X
Y
of elementary
"
!
-polycycles:
p.29/4
Subcase of $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
40/73
p y y
Sporadic elementary
"
!
-polycycles:
p.30/4
Subcase of $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
41/73
p.30/4
Subcase of $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
42/73
The infinite series of elementary
"
!
-polycyclesV V
:
The only elementary infinite
"
!
-polycycles are V
and snub -antiprism.
The infinite series of elementary
"
!
-polycycles
snub0
-antiprisms,0
u
:
p.30/4
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
43/73
VI. Application
to extremalpolycycles
p.31/4
Definition
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- 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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
46/73
n
j
W
extremal components
1 0`
2 0`
`
3 1
4 2
5 3
W
p.34/4
Small extremal $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
47/73
n
j
W
extremal components
6 53
n
7 6i
W
8 83
d
9 103
W
10 123
p.34/4
Small extremal $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
48/73
n
j
W
extremal components
11 153
12 10
i
`
`
`
`
p.34/4
Extremal $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary 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
Extremal $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
Extremal $ -polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
53/73
VII. Application
to non-extendiblepolycycles
p.38/4
Definition
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
56/73
Finite non-extendible polycycles
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary 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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary 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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
59/73
VIII. Application
toface-regular spheres
p.43/4
Euler formula
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
Polycycles of $ -sphere
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
63/73
NO
NO
NO
NO NO p.46/4
Polycycles of $ -sphere
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
64/73
YES NO NO
YES
NO
p.46/4
Polycycles of $ -sphere
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
65/73
The infinite series of elementary
!
"
-polycyclesV V
:
YES NO
NO
p.46/4
$ -sphere
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
67/73
A
!
w
-sphere
w
p.47/4
$ -sphere
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
68/73
The decomposition into elementarypolycycles. p.47/4
$ -sphere
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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
-
8/3/2019 Michel Deza, Mathieu Dutour Sikiric and Mikhail Shtogrin- Elementary polycycles
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