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8/3/2019 Mathieu Dutour, Michel Deza and Mikhail Shtogrin- Polycycles and their boundaries
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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
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I. -polycycles
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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
.
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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
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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.
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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.
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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)
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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
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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.
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II. -polycycles
with given boundary
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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
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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
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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.
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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
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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
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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
.
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P ibl filli
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Possible filling
Let us illustrate the algorithm for the simplest case
.In some cases we can complete the patch directly.
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P ibl filli
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Possible filling
Let us illustrate the algorithm for the simplest case
.In some cases we can complete the patch directly.
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P ibl filli
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Possible filling
Let us illustrate the algorithm for the simplest case
.In some cases we can complete the patch directly.
But in some cases more is needed:
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Diff t ibl ti
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Different possible options
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Different possible options
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Different possible options
or
p.17/4
Different possible options
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Different possible options
or
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Algorithm
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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.
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An example of a search
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An example of a search
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An example of a search
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An example of a search
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An example of a search
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An example of a search
p.19/4
An example of a search
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An example of a search
p.19/4
An example of a search
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An example of a search
p.19/4
An example of a search
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An example of a search
p.19/4
An example of a search
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An example of a search
p.19/4
Another possible search
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Another possible search
p.20/4
Another possible search
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Another possible search
p.20/4
Another possible search
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Another possible search
p.20/4
Another possible search
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Another possible search
p.20/4
Another possible search
8/3/2019 Mathieu Dutour, Michel Deza and Mikhail Shtogrin- Polycycles and their boundaries
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p a
p.20/4
Another possible search
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p
p.20/4
Possible speedups
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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
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III. maps of -gons
with a ring of -gons
p.22/4
The problem
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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
8/3/2019 Mathieu Dutour, Michel Deza and Mikhail Shtogrin- Polycycles and their boundaries
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Main Theorem
Besides the cases
=
and
with
, all such maps
are known;
If
, then the map is
; from now, let
.If
, two possibilities:
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Case
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If
, two possibilities:
and an infinite serie
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Case
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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
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4,
;8,8 10,
;12,10 12,
;10,14
12,
;13,11 12,
;12,12 12,
;12,12 p.27/4
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16,
;14,14 16,
;14,14
20,
;16,16 20,
;16,16 p.28/4
Case
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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
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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
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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
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3,
;9,9 4,
;10,10 8,
;10,18
9,
;19,11 10,
;10,2210,
;14,18
p.32/4
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11,
;20,14 12,
;26,10 12,
;14,22
12,
;14,22 12,
;21,15 13,
;15,23 p.33/4
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14,
;28,12 16,
;30,14
18,
;14,34 18,
;33,15 p.34/4
Known
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6,
;10,20 10,
;34,8
12,
;40,8 12,
;39,9 p.35/4
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12,
;38,10 12,
;38,10
12,
;38,10 p.36/4
Known
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2,
;10,10 6,
;12,24 6,
;14,22
6,
;13,23 6,
;14,22 6,
;12,24 p.37/4
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6,
;15,21 12,
;11,4914,
;58,10
14,
;11,57 14,
;11,57 14,
;11,57 p.38/4
All parameters
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maps
(full.)
maps
(Dode.)
(full.)
(azu.)
?
(azu.)
p.39/4
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IV. Generalizations
p.40/4
Several rings
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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)
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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
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; 44 ; 44
; 60
; 60
; 68
; 68 p.43/4
Two rings of -gons filled by -gons
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; 68
; 68
; 68
; 68
; 76
; 68
p.44/4
Remaining graphs
(azulenoids)
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; 76 ; 80
; 92
; 100 p.45/4
The case = (fullerenes)
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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
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For any
, there exists a map
(with
-ringsof
-gons) and a map
(with
-rings of
-gons)
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Infinite families
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For any
, there exists a map
(with
-ringsof
-gons) and a map
(with
-rings of
-gons)
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Infinite families
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For any
, there exists a map
(with
-ringsof
-gons) and a map
(with
-rings of
-gons)
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Infinite families
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For any
, there exists a map
(with
-ringsof
-gons) and a map
(with
-rings of
-gons)
p.47/4
-valent maps
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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
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