E COLE NORMALE SUPERIEURE

______

A Zo o of ` embeddable

Polytopal Graphs

Michel DEZA

VP GRISHUHKIN LIENS ______

Département de Mathématiques et Informatique

CNRS URA 1327

A Zo o of ` embeddable

Polytopal Graphs

Michel DEZA



VP GRISHUHKIN

LIENS

January

Lab oratoire dInformatique de lEcole Normale Superieure

rue dUlm PARIS Cedex

Tel

Adresse electronique dmiensfr



CEMI Russian Academy of Sciences Moscow



A zo o of l embeddable p olytopal graphs

MDeza

CNRS Ecole Normale Superieure Paris

VPGrishukhin

CEMI Russian Academy of Sciences Moscow

Abstract

A simple graph G V E is called l graph if for some n 2 IN there

1

exists a vertexaddressing of each vertex v of G by a vertex av of the n

cub e H preserving up to the scale the graph distance ie d v v

n G

d av av for all v 2 V We distinguish l graphs b etween skeletons

H 1

n

of a variety of well known classes of p olytop es semiregular regularfaced

zonotop es Delaunay p olytop es of dimension  and several generalizations

of prisms and antiprisms

Introduction

Some notation and prop erties of p olytopal graphs and hypermetrics

Vector representations of l metrics and hypermetrics

1

Regularfaced p olytop es

Regular p olytop es

Semiregular not regular p olytop es

Regularfaced not semiregularp olytop es of dimension

Prismatic graphs

Moscow and Glob e graphs

Stellated k gons

Cup olas

Antiwebs

Capp ed antiprisms towers and fullerenes

regularfaced not semiregular p olyhedra

Zonotop es

Delaunay p olytop es

Small l p olyhedra and examples of p olytopal hypermetrics

1

tembedding in l

1

App endix gures



The authors has b een supp orted by DFGSonderforschungsberich DISKRETE STRUK

TUREN IN DER MATHEMATIK UNIVERSITAT Bielefeld as well as by DIMANETTHE EU

ROPEAN NETWORK ON DISCRETE MATHEMATICS EU Contract ERBCHRXCT

Introduction

A connected simple graph G is called an l graph if its shortest path distance

1

matrix d seen as a metric space d X on the set X of vertices of G is embeddable

G

m m m

isometrically into some l space The l space is IR with l metric

1

1 1

m

X

jx y j x y jjx y jj d

i i l l

1 1

i=1

Equivalently see AsDe G is an l graph i d is embedded isometrically into the

1 G

n

restriction of l on an ncub e up to a certain scale Clearly resp ectively

1

means that G is an isometric subgraph of the ncub e graph resp ectively

of the nhalfcub e graph We denote these embeddings by G H resp ectively

n

1

G H

n

2

m

Any metric subspace A of l is see Dez hypermetric ie it satises

1

X

b b d x x

i j l i j

1

1ij n

P

n

n

for every distinct x x x A b b b ZZ and b

1 2 n 1 n i

i=1

Clearly in the sp ecial case b the ab ove inequality is the classical tri

angle inequality Call the ab ove inequality gonal kgonal resp ectively if

P

jb j k resp ectively Call a graph G hypermet b

i

i

ric gonal kgonal if d is hypermetric resp ectively satises gonal the

G

k gonal inequalities Any k gonal graph is k gonal Dez

but K C is an example of a k gonal which is not k gonal for

2k +1 k +1

any k So for a graph G the following chain of implications holds

G H G H l graph hypermetric k gonal gonal

n n 1

2 1

In this pap er we systematically study well known p olytopal graphs in connec

tion within the ab ove chain of notions in particular lo oking for l graphs within

1

skeletons of p olytop es with a degree of Euclidean regularity These p olytop es in

clude regularfaced p olyhedra semiregular p olytop es zonotop es Delaunay p oly

top es of dimension and some generalizations of prisms and antiprisms In

DeSt one can nd a related study of isometric up to a scale embeddings of

skeletons of innite graphs of Archimedean and Laves plane tilings into cubic

lattices

Here are other instances of embeddings of graphs which are related to our sub ject

well known cubical graphs induced subgraphs of hypercub es and a hypercub e

embedding with a smal l distortion of distances used in computer architecture

isometric or isometric with a small distortion embeddings of graphs into

other than l spaces normed spaces see LLR and the huge bibliography there

1

The following criterion for a graph G to b e an l graph is known Shp

1

DeGr G is an l graph i it is an isometric subgraph of the direct pro duct of

1

halfcub es and Co cktailParty graphs K skeleton of the crossp olytop e

m2 m

Even a go o d p olynomial algorithm for a recognition of l graphs is known DeSp

1

So the raisons detre for this pap er are the following

To classify l graphs within imp ortant classes of graphs Until now this has

1

b een achieved only for strongly regular graphs DeGr Ko o and two classes

of planar mainly non p olyhedral graphs PSC DeTu

To nd b etter criteria and algorithms for the recognition of l graphs within

1

sp ecial classes of graphs

Since an l embedding pro duces a binary matrix we want to lo ok for new

1

co des having in particular symmetries of the original p olytop es Besides l

1

embeddings for example of chains of hexagons b enzenoid chains and some

fullerenes see x give new ways of an enco ding of chemicallyrelevant p olyhedra

and counting molecular parameters dep ending only on graph distances

To confront metrics of skeletons of p olytop es distinguished by their Eu

clidean l prop erties to a larger l metric We remind that an l space is a metric

2 1 p

subspace of an l space for any p For example we shall see b elow how

1

strong l ness o ccurs in zonotop es and interesting p olytop es

1

The somewhat surprising but easy observation that interesting p olyhedra have

l skeletons was one of the origins of this work

1

1

P H if P is one of Platonic solids

10

2

P H if P is one of Voronoi p olyhedra see x

6

1

H if P is one of Delaunay p olyhedra P r ism P y r P

6 3 3 3 3 4

2

1

P H if P is one of chemically imp ortant for main group elements

6

2

co ordination p olyhedra icosahedron P r ism P y r

3 3 3 4

P H if P is one of isozonohedra ie zonohedra with the same rhombic faces

6

see x rhombic do decahedron second Bilinskis rhombic do decahedron

3

rhombic icosahedron and triacontahedron See Cox for names of those p olyhedra

Also skeletons of regular partitions of a plane are scale isometrically em

4

b eddable into the cubic lattice ZZ But in IR interesting non gonal p olytop es

6

app ear for example the cell which is regular and is a Voronoi p olytop e and a

co ordination p olytop e for the ro ot lattice D D

4

4

We are grateful to the SFB University of Bielefeld for their hospitality while

we carried out much of this work We would like to thank Walter Deub er for his

kind invitation interest and supp ort We are grateful to Andrea Walz for drawing

the computer artwork

Some notation prop erties of p olytopal graphs

and hypermetrics

Here we give denitions and notation of graphs and p olytop es that we use b elow

A p olytop e of dimension n is called np olytop e A p olytop e is called a polyhe

dron The p olytop e P is dual of the p olytop e P and P P For a p olytop e

P we denote by GP its skeleton In other words GP is a graph on vertices

of P such that two vertices are adjacent in GP i they are endp oints of an edge

of P

We use the following usual notation of graphs

K is the complete graph on n vertices

n

is the multipartite graph with n parts of sizes k k If n and K

1 n k k

n

1

k p k q we have K the bipartite graph

1 2 pq

If k for all i then we have K the Co cktail Party graph

i n2

n

T n is the triangular graph on vertices the line graph of K

n

2

J n k k n is the Johnson graph in particular J n K J n

n

T n

is the cycle on n vertices i i in this order C C

1 n n i i

n

1

is the path on the m vertices i i in this order P P

1 m m i i

m

1

1

H is the cub egraph H is the halfcub egraph

n n

2

G e is the graph G without its edge e

rG denotes the susp ension of G ie the graph G with a vertex which is not in

G and is adjacent to all vertices of G

m

G r G rrr

z

m

The rst of the following notation of p olytop es are used in Cox

is the nsimplex and G K

n n n+1

is the ncrossp olytop e and G K

n n n2

is the ncub e and G H

n n n

1 1 1

is the nhalf cub e and G H

n n n

2 2 2

ambo is the convex hull of the midp oints of all edges of and Gambo

n n

T n

n

C is a planar ngon and GC C

n n n

1 1

ambo C Note that ambo

2 3 2 3 3 4 4 3 3

2 2

If P is an np olytop e then P y r P is the n pyramid with the base P and

GP y r P rGP The ap ex of P y r P is its vertex which do es not lie in the

nspace spanned by P and is adjacent to all vertices of P An n bipyramid

B P y r P with the nbase P We have

m m m

P P y r P and GP y r P r GP P y r P y r P y r

z

m

P y r P y r C is a pyramid and GP y r rC

n n n n

B P y r P is n bipyramid with the nbase P It has two ap exes which are

2

opp osite with resp ect to the hyperplane spanned by P GB P y r P r GP e

2

where e is the edge connected in r GP the two ap exes

A tcapped p olyhedron is one obtained from a p olyhedron P as follows Cho ose

some t faces of P usually with a maximal number of vertices To each chosen face

add a vertex so that this vertex do es not lie in the plane spanned by the face and is

connected by an edge to each vertex of the face

If P is an n p olytop e i then the direct pro duct P P is an n n

i i 1 2 1 2

p olytop e and GP P GP GP

1 2 1 2

P r ismP P is the prism with the base P

1

P r ism P r ismC

n n

AP r ism is the antiprism with the base C It is a p olyhedron having two

n n

ngon faces n n and all edges i i i i i n with

addition mo dulo n

G G means that the graph G is an isometric subgraph of G

1 2 1 2

Remind that

G H

n

means that G is isometrically embeddable into H with the scale Similarly we

n

1

H if G is embeddable into H with the scale m Also write G

n n

2m

m m m m

P y r P y r B P y r P y r P y r

n n+m n n+m n n+m n

m+3

P r ism B P y r

n n n

n n n

We supp ose that denitions of Voronoi Delaunay and co ordination contact

p olytop es are known to the reader see CoSl for details

Vector representations of l metrics and hypermetrics

All l metrics d on n p oints or l metric spaces d X jX j n form a cone in

1 1

n

dimensional space of all matrices kdi j k i j n This cone is called

2

the cut cone C ut The extreme rays of C ut are cut metrics S for all S X

n n

containing a given p oint of X where S i j if jS fij gj and

otherwise Some of facets of C ut are describ ed by the hypermetric inequalities

n

Moreover for n all facets of C ut are hypermetric For n the cut cone has

n

non hypermetric facets and not each hypermetric inequality determines a facet

The cut cone C ut lies inside a hypermetric cone H y p the cone of all metrics which

n n

P

n

n

satises the hypermetric inequalities for all b Z with b

i

i=1

It is known DGL that H y p is p olyhedral ie it has a nite number of facets

n

and extreme rays A hypermetric lying on an extreme ray is called extreme

Any hypermetric di j on n p oints can b e represented by a generating nsubset

of the set of vertices of a Delaunay p olytop e with di j equal to the squared Eu

clidean distance b etween corresp onding vertices This Delaunay p olytop e is uniquely

constructed by the hypermetric d and its dimension is not greater than n

A Delaunay polytope of a lattice is the convex hull of lattice p oints on the b ound

ary of a lo cally maximal empty from lattice p oints sphere in this lattice Voronoi

proved that there is a nite number of combinatorial types of Delaunay p olytop es

in each dimension

Let P b e the Delaunay p olytop e vertices of which represent a hypermetric di j

i j n and let r b e the radius of the empty sphere circumscrib ed P If we

take the center of P as a new origin then vertices of P are represented by vectors

of the same length r A subset X of vertices of P is called generating if every vertex

of P is an integer ane combination of vertices from X

Let v represents a vertex i X Then we have

i

2 2

di j v v r v v

i j i j

where v v is the inner pro ducts of these vectors If X f ng and v represent

i j

a vertex of P we have

n n

X X

b and b ZZ b v where v

i i i i

i=1 i=1

Every integer combination of v denes a p oint of a lattice whose Delaunay p olytop e

i

is P Since P is inscrib ed in the empty sphere of radius r we have

n

X

2 2

b v r

i i

i=1

P

n

This inequality holds as equality if the vector b v represents a vertex of P

i i

i=1

Expanding this inequality and using the equality we obtain the hypermetric

inequality We see that any vertex of P determines an equality which is satised

by the hypermetric d In particular if n k where k is dimension of P

P P

n n

b b ZZ b v with then there is an ane integral dep endency

i i i i

i=1 i=1

P

n

2

b etween vectors v i X The expansion of the equality b v determines

i i i

i=1

a hypermetric equality which is satised by d This equality is called an mgonal

P

n

jb j equality for even m

i

i=1

Now we present a metho d of an economic recognition of l embeddable metric

1

spaces often used in constructions b elow If d is an l metric then P can b e iso

1

metrically inscrib ed into an mdimensional b ox such that the set of vertices of P is

a subset of all vertices of the b ox and m n k If d is an integral l metric

1

then P can b e isometrically inscrib ed into a cub e and GP is either H or

m m

1

H Below we denote by av f mg the vertex of corresp onding

m m

2

to v

When a graph G is embedded into a hypercub e with a scale then endp oints of

every edge e of G are mapp ed into opp osite vertices of a dimensional cub e This

cub e is spanned by a set N e of mutually orthogonal vectors f i N e In

i

P

f Conversely other words each edge e of G is represented by the vector

i

iN (e)

we can say that each vector f denes a zone Z fe N e ig edges of which

i i

determine a cut of the graph G Each cut corresp onds to a subset S of vertices of G

such that edges of the cut have exactly one vertex in S Hence the indicator vector

of the zone Z has the form S for some S

i i i

It is shown in DeSp that the amount pe e jN e N e j can b e computed

from the distance matrix d Moreover pe e takes only values and Since

2

pe e is an integer it takes only values if is o dd In the case of o dd the

zones are either disjoint or coincide Hence any o dd gives in a sense the same

embedding as an embedding with when all zones are disjoint ab ove we

denoted this embedding by H This is the case of zonotop es see x For

n

P

S with co ecients the zones determine uniquely the expansion d

i G

i

Most of the embeddings with an even are with In particular this is the

1

H For the zone case of this pap er we denoted ab ove this embedding by

n

2

P

1

determine the expansion d S with halfinteger co ecients In the case

G i

i

2

the function pe e takes the values Obviously pe e if and only

if N e N e Hence the equality pe e determines an equivalence relation

on all edges of the considered graph Clearly if pe e and pe e then

pe e Hence we obtain the following criteria of an l embedding of a graph G

1

1

into H the function pe e computed from the distance matrix d should have

n

2

the following prop erties

pe e is nonnegative and integral

the equality pe e determines an equivalence relation

if pe e then pe e pe e for every edge e

If at least one of this conditions is violated then G is not l embeddable But

1

it can b e hypermetric We give criteria for a non l graph to b e hypermetric in the

1

end of this section

We seek l embeddings of graphic metric spaces d X of skeletons of p olytop es

1

with the set of vertices X This means that we seek a representation of p oints of

X by vertices v i X of a Delaunay p olytop e P If n k k dimP

i

then some vectors v can b e linearly and integrally expressed through other vectors

i

Usually one can always nd k anely indep endent vectors which comp ose an

ane lattice base of P This fact is not proved in general but we do not know any

example of a Delaunay p olytop e without a lattice base

So it is clear that if a metric space d X determines a k dimensional Delaunay

p olytop e then it is sucient to nd a representation of k anely indep endent

p oints of X For a graphic metric space d X it is usually not dicult to nd

some dep endencies of p oints of X

We give some congurations corresp onding to dep endencies

The most simple conguration consists of p oints forming a square Four

p oints of a distance space d X comp ose a square if we can lab el the p oints by the

numbers such that d d d d and d d

d d We see that any p oints of the square satisfy a triangle equality

and all p oints satisfy a gonal equality If d is graphic and di i

G

i addition mo dulo then the p oints induce a square cycle of the

graph G

A hexagon dep endency Six p oints of d X comp ose a hexagon if they can

b e lab eled by numbers i i such that di i a the addition mo dulo

and di j a for other pairs of these p oints Note that any p oints of the

hexagon satisfy a gonal equality and all p oints satisfy a gonal equality

We formulate the ab ove as the following

Lemma Let a hypermetric space d X contains i squares and j hexagons Then

deleting i j points of X which cuts al l squares and hexagons we obtain a reduced

hypermetric space d X which uniquely determines the original metric on the set

X

Similarly one can dene other congurations If we nd a conguration we can

delete any p oint of the conguration and seek a representation of other p oints This

metho d of reducing of an embedding problem works very well if the ground graph

of a graphic metric has many squares

Note that if d of Lemma is not hypermetric then d is not hypermetric to o

But if d is not hypermetric it can o ccur that d is hypermetric but d is not Then

the ab ove reduction do es not work

If d is a nondecomp osable path metric of a graph G which is a hypermetric

G

but not l graph then d is an extreme hypermetric and G is an isometric subgraph

1 G

of the Gosset graph G DeGr The Gosset graph has vertices its diameter

21

is and for every its vertex v there is exactly one vertex at distance The vertices

of the Gosset graph can b e lab eled by the pairs ij and ij i j such that

the distances b etween corresp onding vertices are as follows

dij k l jfij gfk l gj dij k l dij k l In particular dij ij

We have the following criterion of a not l metric d to b e hypermetric and

1 G

therefore extreme see DGL

Lemma Let a graphic metric d not be l metric Then d is a hypermetric if

G 1 G

and only if

i the graph G has diameter at most

ii G contains vertices such that the restriction of d on these vertices is one

G

of extreme hypermetrics described by graphs G i of DGL

i

iii if G has diameter then for every vertex v of G there is at most one vertex

at distance from v

The following result was proved rst in CDG

Lemma Let G be an graph Then G is an isometric subgraph of a halved cube

1

ie its scale is if either

i G is planar

or ii G is partite

Pro of

If G is hypermetric then it generates a Delaunay p olytop e PG and contains

an ane basis of PG If G is an graph and its scale then PG is a direct

1

pro duct of halved cub es and crossp olytop es An ane basis of a direct pro duct

is a union of ane bases of the comp onents with one p oint common Any ane

basis of a crossp olytop e contains an ndimensional simplex The skeleton

n

of this simplex is the complete graph K Hence if G is planar the corresp onding

n

direct pro duct PG can contain crossp olytop es only for n The skeletons

n

of co cktail party graphs are isometric subgraphs of a halved cub e if n A

n

1 1

H is an isometric subgraph of the halfcub e H direct pro duct of halved cub es

q n

i

2 2

P

q The case of lemma follows The same pro of is valid for the with n

i

i

case ii since partite graph do es not contain K 2

5

We use the opp ortunity to correct a misprint of the pap er DGL The graph

G b elongs to the class q but not to the class q as it is shown on Fig

18

Regularfaced p olytop es

2

Convex regular p olytop es are dened by induction on dimension In IR they are

regular p olygons A regular np olytop e of dimension n for n is one having

only regular facets and regular vertex gures the convex hull of all midp oints of

edges through a given vertex

A regularfaced np olytop e is one having only regular facets A semiregular n

p olytop e is a regularfaced np olytop e with equivalent vertices ie the group of

symmetries of the p olytop e is transitive on vertices It is quasiregular if moreover

this group is transitive on edges All regularfaced p olytop es are known

Regular p olytop es Platonic solids and semiregular p olytop es Archime

dean solids prisms antiprisms have b een known since antiquity Archimedean

p olyhedra and their dual Catalan p olyhedra were rediscovered during the Renais

sance and Kepler gave them their mo dern names

Semiregular np olytop es were found in by TGosset Mak for n

and BlBl for any n proved that the Gossets list is complete see Cox and

Gr u for the historical account

All regularfaced p olytop es was found by the work of many p eople es

p ecially of NWJohnson and VAZalgaller see for example Ber Zal

KoSu Finally in BlBl the complete list of regularfaced np olytop es is

given

Below we rep ort the status of regularfaced np olytop es see x for not semi

3

regular ones in IR versus l ness of their skeletons By abuse of language we often

1

denote a p olytop e by its skeleton

Regular p olytop es

1

a for n C H and moreover C H including b oth Delaunay

n n 2m m

2

p olytop es C C and b oth Voronoi p olytop es C C

3 4 4 6

1 1

b for n icosahedron H do decahedron H this was proved in Kel

6 10

2 2

in weaker form of the scale embedding

1 1

c G H G K H G K H i K

n n n n+1 n+1 n n2 4m n

2 2m

1

H

4m

2m

In fact l embeddings of and are not unique for n In particular

1 n n

1 1

H for n mo d and K H for n K

n n

2dn2e1 4dn2e2

dn2e 2dn2e

N 1

of the ratio for mo d Those embeddings give the minimal value

dn2e 2m

1 2

any l embedding K H Hence G H for n mo d Also

1 n N n n

2m n+1

1 1

H for n mo d and G H for n G

2dn2e n 4dn2e n

dn2e 2dn2e

mo d

d for n cell is non gonal cell is not gonal see Ass cell is

not gonal see DeGr

Semiregular not regular p olytop es

They b elong to the following Gossets list see for example BlBl we use notation

of Cox

1

a In dimension with the skeleton T H snub cell tetricosa

21 5

2

hedric and o cticosahedric p olytop e last two are undecided

1

In dimension with the skeleton H

21 5

2

In dimension with hypermetric but not l skeleton

21 1

In dimension with hypermetric but not l skeleton

21 1

In dimension skeleton of which is the ro ot graph of all ro ots of the

21

ro ot system E It is not l graph since it contains the skeleton of as an induced

8 1 21

subgraph of diameter and therefore as an isometric subgraph

b In dimension

1

n

H and for even n H P r ism

n+2 +1 n

2

2

1

H for n

4

2

B P y r P r ism

n

n

non gonal for n

1

AP r ism H

n n+1

2

In fact remember that the vertices of AP r ism are i i i i n with edges

n

i i i i i i i i here and b elow the addition is mo dulo n We

1

give embeddings AP r ism H addressing the vertex v to a subset av of

n n+1

2

f n n g for cases b elow i n

n k is o dd

ai fi i i k g ai fn g fi k g ai

n k is even

ai fi i i k g

fn g ai fig if i is o dd

ai

fn g ai fi k g if i is even

n k is even

ai fi i i k g

fn g ai fig if either i k and i is o dd

or k i k and i is even

ai

fn g ai fi k g otherwise

H for n

3

AP r ism

n

non gonal for n

We give in Table the results on l ness of Archimedean p olyhedra and their

1

duals from DeSt

Remark The th Archimedean solid twisted rhombicubo ctahedron

unique new p olyhedron if we weaken the vertex transitivity to the uniqueness of

the vertex gure and its dual are non gonal

1

H for any p olytop e without triangular Remark PSC gives P

m

2

faces and without vertices of degree The truncated icosahedron ab ove provides

an example of a p olytop e without triangular faces but with all vertices of degree

which is non gonal Also non gonal p olyhedra AP r ism M see

4 20

x b elow have only square faces

Table

p olyhedron P emb of P DP emb of P D P

1

truncated tetrahedron non gonal H

7

2

truncated o ctahedron H non gonal

6

1

H truncated cub e non gonal

12

2

cub o ctahedron non gonal H

4

truncated cub o ctahedron H non gonal

9

1

rhombicubo ctahedron non gonal H

10

2

1

snub cub e non gonal H

9

2

1

truncated icosahedron non gonal H

10

2

1

truncated do decahedron non gonal H

26

2

icosido decahedron non gonal H

6

truncated icosido decahedron H non gonal

15

1

rhombicosidodecahedron non gonal H

16

2

1

H snub do decahedron non gonal

15

2

Regularfaced not semiregular np olytop es for n 

Going through the list of those p olytop es given in BlBl we obtain

GP y r K K and GB P y r K e K

n1 122 n2 n1 n+2 211

K

(n+1)2

so b oth these graphs are l graphs Besides for the following two p olytop es

1

we have

P y r icosahedron is not gonal Figf B P y r icosahedron is non gonal its

skeleton contains K P P

5 2 3

1

The union of and P y r where is a facet of H

21 3 3 21 5

2

Finally the set of p olytop es arising from the cell by sp ecial cuts of vertices

is undecided

Prismatic graphs

In this section we group observations on l status of some generalizations and rela

1

tives of prisms and antiprisms

Moscow and Glob e graphs

m

Call M P C m n Moscow graph since its pathmetric is called

m n

n

in computer geometry Moscow metric This graph is sometimes called annular

city street graph and this metric also app ears under the name radardiscrimination

distance P r ism is the case m of Moscow graph

n

1 1

H since P H so P H and Clearly P C

n+2m2 m m1 m 2m2 m n

2 2

1

C H Also P C H since C H

n n m 2k k +m1 2k k

2

The Moscow graph is p olyhedral since it is planar and connected This p oly

hedron can b e seen as a tower of m copies of P r ism put consecutively one on

n

the top of other

m m1

Call the skeleton of the dual M the Globe graph and denote it M It can

n n

m1

b e seen as a capp ed Moscow graph M P C and as melongated

m1 n

n

bipyramid For m it is the usual B P y r and for m it is the usual

n

elongated bipyramid of the list of regularfaced p olyhedra are the

2

cases n of M

n

Prop osition

1

H if n

2m+2

m

2

M

n

is non gonal if n

Pro of Let ij i m j n m b e the mn vertices of the Glob e

m

graph M such that all its edges are either on meridian paths P

01j(m1j )m

n

j n or on parallel cycles C i m For n an embedding

i1in

1

H is given by in

2m+2

2

a am f m g aij f j g ft ZZ t i g

For n such an embedding is given by

a am f m g aij fj j mo d g ft ZZ t

i g

We show non gonality of the Glob e graph by exhibiting vertices x y a b c

such that

i dx y da b da c db c dx a dx b dx c dy a

dy b dy c

They are

for m for m

m dme m for m

i ab ove take the form

for m for m for m

mdmemmdme m dmemm

m dme m if n m dme m m d me

m dme m m m dme m if n 2

m m

Finally we consider the graph M M with the deleted vertex m ie

n n

capp ed Moscow graph It is the usual pyramid for m Clearly it is the skeleton

of a selfdual p olyhedron

1

H if n

2m+n2

m

2

Corollary M

n

is non gonal for n

m m

Pro of In fact M is an isometric subgraph of M if n But the cycle

n n

C is not an isometric subgraph the p olar way via is shorter than

(m1)1(m1)n

within this cycle i bnc m So embeddings for n come by

deleting the vertex m For n an embedding is given by a aij

fj j mo d g ft ZZ t i g for i m j For n

one can nd see Figa for n p oints violating gonal inequality 2

Another than Moscow graph generalization of P r ism C C is the Lee

n 2 n

Q

t

C its path metric is the Lee distance graph

n

i=1

i

P

t

d x y minjx y j n jx y j

i i i i i

i=1

Lee

used in the co ding theory a discrete analog of elliptic metric Remind that H

4

C C since C H

4 4 4 2

Q

t

1

n

H and moreover H Clearly C if all n are even where n

n n i

i=1 i

2

2

P

t

n

i

i=1

Q

t

P H it is a grid graph with usual grid distance ie l distance

n nt 1

i=1

i

P Q

t t

1

H it is the Hamming graph with Hamming metric jx y j K

nt i i n

i=1 i=1 i

2

jf i t x y gj

i i

Stellated k gons

0 0 0

with the additional edges of the k Stel lated k gon S tel is the k cycle C

k 11 22 k k

cycle C So S tel is C with a triangle on each edge Besides S tel is AP r ism

12k k k k k

0 0 0

Also it is an isometric subgraph of skele without the edges of the k cycle C

1 2 k

tons of many imp ortant p olyhedra for example of icosahedron and others with

vertices convex deltahedrons for k of cub o ctahedron and snub cub e for

k of icosido decahedron and snub do decahedron for k of Archimedean tiling

for k For n it is used as a well known symbol Red Star Star

of David Muslim Star resp ectively

Denote by S un the isometric subgraph of S tel obtained by deleting vertices

2k 2k

i i k It is an isometric subgraph for example of truncated tetrahe

2

dron truncated cub e of Archimedean tiling for k resp ectively

1

Prop osition i S tel H

k 2k

2

1

ii S un H

2k 3k

2

Pro of In fact let i i i k denote vertices For o dd k m an

embedding is given by ai fi i i m g ai fi i i m m

ig for i k addition is mo dulo k except for m i

For even k m we dene ai inductively We take am f k g Then

ai ai with two largest elements deleted if i m

ai ai with two smallest elements deleted if m i k

am f k k mg am f k k m g

ai fk ig ai fk i g with two largest elements deleted if

i m

ai fk ig ai fk i g with two smallest elements deleted if

m i k

Cup olas

Denote by C up the graph with vertices i i i i n and edges on the

n

0 00 0 00 0 00

for all and on the paths P inner cycle C on the outer cycles C

i ii 1n 1 1 n n

i n So C up is the skeleton of a p olyhedron for n they are

n

triangular square and p entagonal cup olas the p olyhedra in the list of

regular faced p olyhedra

1

Prop osition i C up H for n

n 2n

2

ii C up and C up n are non gonal

3

n

Pro of In fact C up contains a non gonal isometric subgraph v see

3

denitions in x on p oints

1

H We take an embed We give the following explicit embedding of C up

2n n

2

1

ding of the inner cycle C H Let the vertex i is mapp ed in this embedding

1n n

2

into an subset ai of an nset V Let a and V f ng Then

we map the vertices i and i into the sets ai fi i g and ai fi i g

resp ectively the addition is by mo d n Note that the shortest path b etween non

adjacent vertices of the outer cycle go es through the inner cycle Hence it is not

dicult to verify that the obtained embedding is isometric for n

C up contains K as an isometric subgraph for n 2

23

n

Antiwebs

n

k

c is the graph with the vertices n and An antiweb AW for k b

n

2

i i i i k mo d n Here i j denotes that the vertex i is adjacent

to the vertex j It is a common generalization of the following p olyhedral graphs

1

1

n

AW C H and H if n is even

n n

n

2

2

1

2

n

AW GAP r ism H if n is even

n

n

2

2

1

bn2c

AW K G H

n n1 n

n

2

n

1

1

2

n n

G K AW H for some m if n is even

n

2 4m

2m

2 2

In the next prop osition we show that all other antiwebs are not l graphs It is

1

n n+2

k

c k b c easy to check that AW has diameter i b

n

4 2

2 2

Prop osition i AW is non gonal for odd n AW is not gonal

n 11

k

ii AW is non gonal in the fol lowing cases

n

n n+1

e k a if d

4 2

b if n k for k and if n k for k

n3

k

iii AW is not l embeddable if either k and n or k and

1

n

2

3 4

therefore n In particular AW and AW are not gonal

13 16

2 3 3

iv AW AW and AW are extreme hypermetrics

9 12 14

2

Pro of i We show explicitly p oints a b c x y of AW where the gonal

n

inequality with b b b b b is violated

a b c x y

If n t take the p oints t t t t

If n t take the p oints t t t t

2

Similarly for AW we have the following p oints violating a

11

gonal inequality

k

ii a In fact any A with k as in Prop osition is of diameter so any its

n

induced subgraph of diameter is isometric Only connected graphs on vertices

which are non gonal are K K K and K P P with P P

23 5 3 5 2 3 2 3

k

Hence the only way for AW of diameter to b e non gonal is to contain one of

n

these graphs on vertices But any antiweb is K free graph Hence the rst

13

ab ove subgraphs K and K K are excluded On the other hand we will give

23 5 3

subgraph K P P namely the complement of

5 2 3

P P for any i with n i k n i and

12k i4k 2i1 k in+1k

i k

n+1 n3

So it will cover all cases n k n k ie d e k b c

4 2

considered in Prop osition

k

Now P AW since k k i k i k k i

12k i4k 2i1

n

b ecause i k and k i n k i k n

k

P AW since k i k i k n k n But k i n k

k in+1k

n

k

k i k i in AW

n

ii b If n k take the p oints k k k k and if

n k take the p oints k k k k

3

iii For k consider the subgraph of AW induced by the p oints i i

n

Here the p oint is adjacent to all other p oints Since n the p oint is

not adjacent to the p oints Finally we have that the induced subgraph is

k

rB For k consider the subgraph of AW induced by the following p oints

8

n

n3

k k k k Since k this graph is rK P rH

6 4 2

2

where P P The graphs B and H are the graphs on Figs and

4 31129 8 2

of DGL Since rB G and rH G are graphs from the list of the

8 4 2 2

k

extreme hypermetrics on p oints we have that AW is not l embeddable

1

n

3

The following p oints violate a gonal inequality for AW and

13

4

for AW

16

2

iv The graph AW has hexagons and

9

determining gonal equalities Any two of them are linearly indep endent Hence

representations of two vertices say and are determined by representations of

other vertices If we delete the vertices and we obtain the extreme hypermetric

2

G from Fig of DGL Since AW has diameter it is a subgraph of the

19

9

Shlai graph For example the following lab elings the vertices by the pairs

2

gives an embedding A prop os AW e is a

9

p olyhedral graph see Figc

3

Similarly we nd that the vertices i i of the graph AW induce the

12

3

extreme hypermetric graph G rB The following lab eling shows that AW is

4 8

12

an extreme hypermetric graph

and

3

The graph AW has diameter and the set of its vertices consists of pairs of

14

vertices at distance The vertices i i induce the extreme hypermetric

3

graphs G rB Hence AW is not l graph But the following lab eling shows

4 8 1

14

that it is extreme hypermetric

and i i 2

Capp ed antiprisms towers and fullerenes

2 2

We consider here an antiprism analog of capp ed prisms M M of x Take

n n

1

at rst AP r ism The icapp ed H i In fact they are

3 3 3 i+4

2

isometric subgraphs of omnicapp ed dual truncated Denote the vertices

3 3

of by i i where i and all i i are not edges and the caps of faces

3

by t and the caps of resp ectively opp osite faces

by t t Put then all ai fi g ai f g ai at ft t g

1

at f g ftg f tg The dual icapp ed H for i

3 i+6

2

but it is non gonal for i

From now on we consider icapp ed AP r ism n only for i ie only

n

ncycles are capp ed Denote by iAP r ism icapp ed AP r ism for i Remind

n n

1

H for n and dual AP r ism is non gonal if n x that AP r ism

n+1 n n

2

2

it can b e seen as a half of the Shrikhande The next case is AP r ism AW

4

8

graph iAP r ism is an extreme hypermetric for i see x and Fig they

4

are of the list of regularfaced p olyhedra see their duals on Fig a

b

1 1

iAP r ism H for i Its dual do decahedron H for i but

5 6 10

2 2

it is non gonal for i Fig c

1

AP r ism H for n but non gonal for n In fact remind

n n+1

2

that vertices of AP r ism are i i for i i n and edges are pairs i i i i

n

i i i i Let b e the cap of C Put then a ai fi i g

1n

ai fi i i n g Clearly for n it will give a scale isometric

embedding into n cub e For n dual AP r ism AP r ism and their

n n

duals are not gonal see Figde for case n

Consider now towers generalizing gyro elongation of antiprisms in a way similar

k

to how the Moscow graph in x generalizes elongation of prisms Denote by T ow

n

the p olyhedron dened by adjoining consecutively k copies of AP r ism one on the

n

2

top of another It is easy to check that T ow is non gonal for n see Fig fg

n

2

for T ow ie gyro elongated it app ears also on Fig in Gr u and its

3

3

k k

dual It implies that T ow and capp ed T ow are non gonal for k Remark

n 5

k

similarity of T ow with p olar zonotop es PZ mentioned in x See also Fige for

m

5

non gonal Ko esters graph Ko e similar to AP r ism AP r ism and to PZ

5 5

5

it is regular planar critical graph ie its chromatic number is but every its

prop er subgraph has a coloring

k

Dual T ow is a ful lerene ie a simple p olyhedron with m vertices

5

p entagonal and m hexagonal faces F for m k Let us call it strained

20+2m

most antiaromatic least stable in chemical terms since it has as opp osite to

preferable fullerenes the maximum number of abutting pairs of p entagonal faces

Except the case k do decahedron the strained fullerene F is non gonal

10(k +1)

for k it contains the graph in Fig b for the case k see Fig c Its dual

2

also non gonal since it contains T ow depicted on Fig a

5

We give b elow some observations on l status of several fullerenes In particular

1

small fullerenes F F F are of sp ecial interest for chemistry b esides b eing

24 26 28

carb on cages they are used Wel p and pp on icelike clathrate

hydrates as space colling p olyhedra There are Bal

fullerenes F for m resp ectively F is the

20+2m 20

do decahedron the unique F is the dual of AP r ism it and F are on Fig

24 6

24

ed the unique F and its dual are on Fig ge F T T is its group of

26 28 d d

symmetry and its dual are on Fig dh F D is on Fig c The strained

30 5h

fullerenes F for m are F D F D F D F D F D

20+2m 30 5h 40 5d 50 5h 60 5d 70 5h

with this symmetry it is unique for m and one of two for m

resp ectively The truncated icosahedron with its icosahedral symmetry is dened as

F I As we checked all ab ove fullerenes and their duals are non gonal except

60 h

1 1 1 1 1

F H F H F H F T H F I H F I

20 10 26 12 6 d 7 h 10 60 h

20 28 60

2 2 2 2 2

1

admits embedding in H see DeSp and x here

20

2

Using also the algorithm of DeSp one can see that the unique preferable ie

without abutting pairs of p entagons F is not l graph This F is second one

70 1 70

with the symmetry D and contains bowl ie the graph consisting of a p entagon

5h

1

H see Figf Also the graph consisting surrounded by hexagons b owl

15

2

of a p entagon surrounded by p entagons or by p entagons and nonadjacent

1 1

hexagons is embeddable into H H resp ectively a hexagon surrounded

10 12

2 2

1

by p entagons resp hexagons embeds into H see Fig g resp into H

12 9

2

regularfaced not semiregular p olyhedra

We use names and numbers of those p olyhedra from Ber where they are given as

of the list of all regularfaced p olyhedra in Zal they numbered

by In b oth references they are given as same appropriate joins of basic

p olyhedra M M See also KoSu with nice pictures of those p olyhedra

1 28

basic regularfaced p olyhedra all but M are given on Fig include regular ones

22

M M pyramids M M cup olas M M The Archimedean p olyhedra M

1 15 2 3 4 6 i

denoted as in Table of x for i f g

resp ectively M M which are tridiminished parabidiminished R I D o M which

13 14 7

is tridiminished I co p entagonal rotunda M and others whose complicate names

9

reect their high individuality In this section we denote icosahedron do decahedron

and rhombicosidodecahedron by I co D o and R I D o resp ectively Remark that

remaining Archimedean p olyhedra of Table are in this notation C up C up

3 3

M M R I D o M C up C up P r ism C up

9 9 14 5 4 8 4

Lemma i M is non gonal for n

n

ii M snub square antiprism is not gonal

28

iii M sphenocorona is extreme hypermetric

22

iv Remaining polyhedra M have l skeletons

n 1

v M has l skeleton for n and it is non

1

n

gonal for remaining polyhedra M

n

Pro of iii are clear from Fig iii is proved in the pro of of the next lemma

1

H l p olyhedra are M M see x M M and M M see x M

4 1 1 15 16 18 26 27 2

2

1 1 1

M H and M H M H see xx the skeleton of M is an

3 5 5 8 6 10 7

2 2 2

1 1

isometric subgraph of l skeleton of I co M H Finally M H

1 7 6 9 11

2 2

1

where M H 2

9 13

2

Lemma Skeletons of the fol lowing polyhedra are extreme hypermetrics

AP r ism P y r gyroelongated P y r capped AP r ism

4 4 4 4

P y r AP r ism P y r gyroelongated B P y r capped AP r ism

4 4 4 4 4

P r ism P y r triaugmented P r ism

3 4 3

M sphenocorona

22

M P y r augmented sphenocorona

22 4

Pro of The skeletons of the p olyhedra and have each vertices and

diameter The skeleton of has vertices diameter and contains the

skeleton of as an isometric subgraph These graphs contain the induced

extreme hypermetrics G and G of DGL Fig This implies that these

24 25

skeletons are not l graphs

1

We give explicit embeddings of skeletons of and into the Schlai graph

G and the skeleton of into the Gosset graph G it will show that these

21 21

skeletons are extreme hypermetrics

Recall that the vertices of AP r ism are i i i i with the edges i i

4

i i i i and i i here and b elow additions are mo d Let b e

the new vertex of the cap of the capp ed antiprism and b e the second cap

of the capp ed antiprism Then the lab els of vertices by pairs are as follows

ai i i i a a a a a

a

Similarly let i i i i b e the vertices of the prism of with edges

ii and let i b e the ap ex of the P y r on the square face of P r ism opp osite to the

4 3

edge i i Then the lab els are ai i ai i ai i i i

The skeleton of has vertices and diameter it contains the induced

extreme hypermetric G It is capp ed AP r ism with an additional edge b etween

25 5

two nonadjacent vertices of the noncapp ed cycle C Let as ab ove the vertices of

5

AP r ism are i i i such that all vertices i are adjacent to the ap ex of the

5

cap and the vertices and are adjacent We lab el the vertices by pairs as follows

ai i i i Since the vertices and are at distance from the

vertices and resp ectively we have a a and a

The other two adjacent vertices obtain the lab els a and a

The skeleton of is G without the vertex which is the ap ex of the

cap P y r The lab eling of G is induced by the ab ove lab eling of G

4

since G is an isometric subgraph of G 2

Prop osition Besides non gonal and above hypermetrics

remaining polyhedra consist of non gonal ones and with

l skeletons except except

1

Corollary l status of convex deltahedra is as fol lows

1

1 1 1 1

H B P y r H H Icosahedron H i

3 3 5 3 4 6 3

2 2 2 2

ii B P y r and M snub dispensoid are non gonal

5 25

iii are extreme hypermetrics

Remark that all dual truncated P where P is a Platonic solid are simplicial and

1

have l skeletons They H with n for P b eing

1 n 3 3 3

2

I co D o resp ectively Between other simplicial Catalan p olyhedra namely P r ism

n

and dual truncated Q Q b eing cub o ctahedron or icosido decahedron only P r ism

n

n are gonal b etween remaining Catalan p olyhedra including AP r ism

n

n only dual Q are gonal

Sketch of the pro of of Prop osition

a p olyhedra

They are all p ossible adjoinings of C up M to gonal faces of two of them

5 6

M and M see Fig Also Archimedean R I D o M M

13 14 14 6

1

M M H One can check that slight mo dications of this embedding

13 6 16

2

1

pro duce an embedding into the same H of M and all p olyhedra without

16 14

2

M M M M M M M M The faces M M

6 14 6 13 6 6 13 6 14 6

remaining p olyhedra are non gonal see Fig for M Slight mo dications of

13

these p oints give due to symmetries forbidden p oints congurations for others

b non gonality

For p olyhedra vertices violating

gonality were found ad ho c M for i see Fig

i

see Figb for

Denote by v k m resp ectively by e k m the graph consisting

of a vertex resp ectively of an edge surrounded by cycles C C C C for

3 2k 3 2m

p ositive integers k m in this order each two consecutive cycles intersect exactly

in the edge incident adjacent to the original vertex resp ectively to the original

edge It is easy to check that b oth the families of the ab ove graphs are not gonal

Now v M anticuboctahedr on

4

v

e M

11

e M

12

2

K P P M M so non gonal see x

5 2 3 25

5

is easy to check Remaining p olyhedra have many vertices they are to o

cumbersome for to present them on gures So we leave them to the reader and give

here only a helpful clarication of them in terms given in Remark b elow

are orthobiC up M is gyrobiC up

5 9 5

are elongated orthobiC up M

5 9

are elongated gyrobiC up C up M

4 5 9

are gyro elongated C up M is gyro elongated biC up

4 9 5

C up M C up M and C up P M C up

5 9 5 9 5 10 9 5

P M

9 9

c l embeddings

1

For p olyhedra l embeddings were found ad

1

ho c Now

M C up M C up M

5 4 6 5 9

I co

1 1

dual truncated D o H H

16 7

2 2

1

dual truncated I co H

10

2

More precisely

1

H since

6

2

AP r ism AP r ism P y r P y r AP r ism P y r I co

5 5 5 5 5 5

M M P y r M P y r I co and M AP r ism I co

7 7 5 7 5 7 5

1

H

6

2

1 1

H since D o dual truncated I co H

10 10

2 2

D o D o P y r P y r D o P y r dual truncated I co

5 5 5

D o P y r D o P y r dual truncated I co

5 5

We have

1 1 1

H M P y r H B P y r H M P y r

4 3 5 5 3 4 2 4

2 2 2

1 1 1

2 2 2

M H M H M H

5 6 7

3 4 5

2 2 2

2 2

1 1

M H M H

6 6

3 4

2 2

See x for those embeddings

Now we present embeddings ad ho c Between them

1 1

C up P r ism H C up P r ism H

4 8 10 5 10 12

2 2

1 1

C up AP r ism H C up H

5 10 11 4 8

2 2

1

M P r ism H

9 10 13

2

1

An embedding H is given on Figa

8

2

others are obtained by capping

1

biaugmented P r ism ie capp ed on square faces H bi

3 5

2

1

augmented P r ism on two nonadjacent square faces H parabi

5 7

2

augmented metabiaugmented P r ism on opp osite or nonopp osite nonadjacent

6

1

H If we relax denitions of by p ermitting to square faces

8

2

cap adjacent square faces then we get nonconvex p olyhedra having same l status

1

of the skeletons as original ones Elongated p olyhedra are easy see

Remark We leave and ab ove capp ed p olyhedra to the reader 2

Remark For a p olyhedron P following for example pp of Ber

call P P P P P P r ism P AP r ism P P r ism P P P r ism P

P AP r ism P resp ectively orthobiP gyrobiP elongatedP gyro elongatedP

elongated orthobiP elongated gyrobiP gyro elongated biP Here ortho gyro

resp ectively means that two solids are joined together such that one of two bases is

the orthogonal pro jection of is turned relative to resp ectively the other Prisms

and antiprisms ab ove have an appropriate base and are adjoined by it List of

regularfaced p olyhedra contains orthobiP and elongated orthobiP for exactly

P M i it turns out that b oth have l skeletons if i and

i 1

b oth are non gonal otherwise All gyrobiP gyro elongated biP and elongated

gyrobiP in the list are non gonal Between gyro elongated p olyhedra of the list

two P y r AP r ism and C up AP r ism have l skeletons and two

5 5 5 10 1

P y r AP r ism and P y r AP r ism P y r are extreme hyper

4 4 4 4 4

metrics Clearly elongated P has l skeleton i P has We have see Fig that

1

1

C up C up is non gonal while C up H and elongated

4 4 4 8

2

C up P r ism C up is not gonal see Remark in x while elongated

4 8 4

1

rhombicubo ctahedron H

10

2

Remark duals of regularfaced p olyhedra

Examples of non gonal ones are duals of cup olas M M

4 6

gyrobifastigium M i

i

M is even strongly non gonal its skeleton as well as K contains

23

20

vertices x y a b c with each of x y b eing on the same shortest path b etween vertices

of the pairs a b a c b c b oth M and K have only square faces M has

23

20 28

only p entagonal faces Other ab ove M have more than one type of faces for

i

example M dual dispheno cingulum has only square and p entagonal faces

24

Examples of l graphs are duals of l graphs and non

1 1

1 1

H M H see Fig gh gonal M M M

10 8 25 23

25 23

2 2

3 2

M Wel p together with a hedron lls IR Duals of are M

25 n

for n see x

1 1

Also the duals of the extreme hypermetrics H H resp ectively

8 9

2 2

see Fig while duals of M are non gonal

22

It is interesting to characterize pairs P P of dual p olytop es having b oth l

1

m m

skeletons Examples M M for n f g icosahe

n n n n

n n

dron do decahedron icapp ed its dual for i icapp ed its dual for

3 3

m1

i P C M for n f g notation are from x

m n

n

Remark Besides ful lerenes F see x ma jority of chemically rele

20+2m

vant p olyhedra ie most frequent ones as arrangement of nearest neighbours in

crystals molecules or ions are regularfaced They include deltahedra see Fig

in SHDC and Corollary ab ove P r ism also its dual and its augmentations

3

P y r cub o ctahedron and capp ed see x antiprisms i

3 4

capp ed AP r ism for n and i capp ed AP r ism icosahedron

n 5

capp ed AP r ism dual of the fullerene F see x Regularfaced p olyhedra are

6 24

also used in large chemical literature on metal clusters see for example DDMP

considering see it on Fig b

Zonotop es

Let e i n b e n mutually orthogonal vectors of a same length Let eS

i

P

e for S N f ng Then the convex hull of the vectors eS for all

i

iS

n

S N is the ncub e The p oints eS are vertices of If we pro ject onto

n n n

a k dimensional space for k n we obtain a zonotop e Z The zonotope Z has

n n

the following three equivalent characterizations it is a pro jection of an ncub e

a Minkowski sum of line segments a p olytop e having only centrally symmetric

faces

Let v b e the pro jection of e Without loss of generality we may assume that

i i

v for all i N Denote the zonotop e Z also by Z v v v Z v i N

i n 1 2 n i

P

v is the pro jection of the vertex eS Let all the vectors v The p oint v S

i i

iS

go out from an origin Then the origin is the p oint v We can take any p oint v S

as a new origin This is equivalent to a change of signs of the vectors v for i S

i

With the origin in v S the zonotop e Z takes the form Z v i S v i N S

n i i

and the p oint v T is transformed into the p oint v S T

The p oint v S is a vertex of Z not for every S There is the following simple

n

criterion when a set S determines a vertex of Z Since every vertex is an extreme

n

p oint for every vertex v S of Z there is a k vector c such that cv S cx for all

n

x Z x v S Here cx is the inner pro duct of vectors c and x In particular

n

this implies that cv for all i S and cv for all i N S In other words

i i

the k dimensional hyperplane fx cx g supporting the vertex v S separates all

vectors v for i S from all other vectors

i

It is easy to see that two vectors v and v are separated by a hyperplane if and

only if they have distinct directions ie v v for some real If v and

k

v have a same direction then Z v i N Z v v i N fj k g where

j i i

k

v v v Hence without loss of generality we may supp ose that all vectors v

k j i

k

have distinct directions

We denote the family of all subsets S determining the vertices of Z by S If

n n

the origin is the vertex v then all vectors lie in a halfspace determined by the

hyperplane supp orting v Without loss of generality we can supp ose that the

origin is a vertex of Z ie S Let V N b e such that v fig v is a vertex

n n i

for all i V Then Z lies in the conic hull of v for i V

n i

Lemma S V for every S S

n

Pro of Let S S and let c b e a vector such that cv for k S and cv

n k i

P

k v for i N S Every v lies in the cone generated by v i V ie v

i i k i k

iV

with k If S V then the inequality cv for all i V implies the

i i

inequality cv for k S a contradiction 2

k

Prop osition The skeleton GZ of any zonotope Z is isometrically embeddable

n n

into the skeleton H of of whose projection Z is n is the diameter of GZ

n n n n

Pro of We prove that the natural map v S eS for S S determines an

n

l embedding of the skeleton of Z into H For this end it is sucient to prove

1 n n

that the distance b etween vertices v S and v T is equal to the symmetric dierence

jS T j We pro ceed by induction on m jS T j

The assertion is trivially true for m By the construction of Z each its

n

edge which connects the vertex v S with another vertex is parallel to the vector v

i

for some i N

Supp ose there is an edge which connects v S with v S and is parallel to v

i

for i S T Then either S S fig or S S fig dep ending on i T or

i S resp ectively Clearly in b oth the cases jS T j m and we can apply the

induction step Hence for to prove the prop osition it is sucient to prove that v S

is incident to an edge parallel to v for i S T But this is implied by Lemma if

i

we take v S as a new origin 2

Komei Fukuda kindly p ermitted to us to see and to refer preliminary version

of Fuk containing within his treatment of oriented matroid a nice example

of a centrally symmetric but non zonohedral p olyhedron F with GF H The

9

Fukudas p olyhedron contains vertices hexagonal and square faces Fukuda

constructed it in the dual form as a nonlinear nonPappus extension of an oriented

rank matroid on p oints As a linear extension of the same matroid he obtain

a dual zonohedron Z Z has vertices hexagonal and square faces

9 9

Comparing Fukudas picture for duals of Z and F we realize that F comes from

9

Z by the following general construction

9

0 0

b e two opp osite hexagonal faces of a centrally Let C C and C C

16 1 6

symmetric p olyhedron P and are pairs of antipo dal vertices of

P Denote by QP the p olyhedron obtained from P by adding two new vertices v

v and edges v v v v v v Clearly GP is centrally

symmetric v and v are antipo dal and GP GQP ie the skeleton of P is an

isometric subgraph of the skeleton of QP If P H it is so if P is a zonotop e

n

then QP H also In fact let a a fij g a fik g Then

n

av fig is uniquely determined No other edge v u is p ossible since otherwise

au fitg for t j k and we get a contradiction with C b eing a hexagonal face

m m1

Dene by induction Q P QQ P Let E l D o denote the elongated do

decahedron Using Remark b elow one can check that combinatorially QP r ism

6

2 2 4

rhombic do decahedron Q P r ism Q E l D o rhombic icosahedron Q

6 4

truncated triacontahedron It will b e interesting to see whether Fukudas

3

p olyhedron is minimal for parameters of nonzonohedral QP obtained from a zono

hedron P

Another similar op eration is as follows Select two opp osite mfaces C C

12m

0 0

Take a path P of length m connecting the and C C

1i i m+2

1 (2m)

1 m2

vertices and m of C Add new edges i k k m Make the

k

same op eration with the face C

Let P denote the rhombic do decahedron with a deleted vertex of degree ie

P is P r ism with a new vertex connected to nonadjacent vertices of a cycle

6

Clearly P is nonzonohedral p olyhedron and GP H for n probably P is

n

minimal for this prop erty Also GP is one of the four smallest nonHamiltonian

p olyhedral graphs realizable as Delaunay tesselations Dil Fig and A

1 1

prop os graphs of Fig of Dil are non gonal except c H a H

8 6

2 2

1

b H b H

7 4

2

Remark Examples of embeddings of zonotop es into hypercub es are

of Archimedean and their dual Catalan p olyhedra are zonohedra

truncated H truncated cub o ctahedron H truncated icosido decahedron

3 6 9

H dual cub o ctahedron rhombic do decahedron H dual icosido decahe

15 4

dron triacontahedron H Between their duals only the dual of the rst one

6

has l skeleton

1

All combinatorially Voronoi p olyhedra are zonohedra

cub eH rhombic do decahedron H P r ism H elongated do decahe

3 4 6 4

dron H truncated H

5 3 6

All golden isozonohedra of Coxeter zonohedra with all faces b eing rhombic

with the diagonals in golden prop ortion are

types of hexahedrons equivalent to H rhombic do decahedron H

3 3 4

rhombic icosahedron H triacontahedron H

5 6

Some innite families of zonotop es are

P r ism H H

2m m+1 m m

p olar zonohedra PZ H rhombic do decahedron rhombic icosahedron

m m 3

for m resp ectively see Cox PZ is non gonal for m see Fige

m

for m

An mdimensional p ermutahedron the Voronoi p olytop e of the lattice A

m

It is C truncated for m resp ectively H

m+1

6 3

2

Remark For an isometric mvertex subgraph G of an ncub e which is not

n

an isometric subgraph of H denote by G the subgraph of H induced by m

n1 n

remaining vertices If G is the skeleton of a zonotop e Z H then G can b e

n

n1

a G if m examples are and triacontahedron

n1

b an isometric subgraph of H for example G C for rhombic icosahedron

n 10

and G C P P for elongated do decahedron see Fig a

110 1113125 61381410

c or not isometric subgraph of a hypercub e for example G K for rhombic

1

do decahedron G K for P r ism and G is a vertex non isometric subgraph of

2 6

H for truncated

6 3

Remark See DeSt for similar isometric embeddings of skeletons of in

nite zonohedra plane tilings into cubic lattices ZZ For example regular hexagonal

n

tiling and dual Archimedean tiling are embeddable into ZZ Archimedean

3

tiling is embeddable into ZZ Penrose ap erio dic rhombic tiling is embed

6

dable into ZZ

5

Delaunay p olytop es

Here we consider Delaunay p olytop es of dimension at most and some op erations

on Delaunay p olytop es see a denition of a Delaunay p olytop e in x

1

H and C H For n Delaunay p olytop es have the skeletons C

3 4 2 3

2

For n skeletons of all Delaunay p olytop es are also l graphs

1

1 1 1 1

H H H P y r H P r ism H

3 3 3 3 3 4 4 4 3 5

2 2 2 2

All types of Delaunay p olytop es are given in ErRy The i i

and the letters ABC b elow are the notation from ErRy Elsewhere than

this section means the number of a p olyhedron in the list of regularfaced

p olyhedra We get the following prop osition by direct check

Prop osition H

4 1 3 4

1

P H for the fol lowing P

4

2

P y r P y r with G K BP y r with GB K

4 2211 3 2221

1

CB P y r H

3 4 4

2

1

P H for the fol lowing P

5

2

P y r P y r P r ism P r ism

3 4 3 1 3 3

with G T

1

P H for the fol lowing P

6

2

P y r P r ismP y r P r ism

2 2 1 4 4 1 3 3

A with GA K the cyclic polytope

6

1

H for P P r ism P r ismP r ism P

7 1 3 3

2

the fol lowing P are non gonal

B P y r P r ism P y r B P y r

3 3 3

Pro of Recall that a graph G is non gonal if K K G or K P P G

5 3 5 2 3

with P P Hence ab ove is implied by that the skeletons of P y r

2 3 3

and B P y r each contains the isometric subgraph K K The skeletons

3 5 3

of and B P y r P r ism contain the isometric subgraph K P P All

3 5 2 3

embeddings in ab ove except either are trivial or come from the

direct pro duct construction b elow The skeleton Gk of the four p olytop es k

in for k are isometric subgraphs of the triangular graph T In fact

G T K G T v v G T v and G T

3

Here T K is the graph T where any mutually adjacent vertices are deleted

3

T v v is T where any two nonadjacent vertices are deleted and T v

is T where a vertex is deleted vertices are deleted with incident edges In other

words the vertices of the p olytop es k k are lab eled by pairs ij

i j For example P y r P r ism has the lab els and i i i

3

2

From this embedding of P y r P r ism we obtain an embedding of P y r P r ism if

3 3

we lab el the new ap ex by the set Remark that is combinatorially equivalent

to a semiregular p olytop e ambo in the Coxeters notation 2

4 21

Also P y r is the graph of the unit cell of the b o dycentered orthorhombic

3

crystal system the graph of all other systems are also not gonal except simple

1

one H and endface centered P y r P y r H See DeDeGr

3 3 4 3 4 6

2

for more detail on it and other chemical applications

It is well known that the direct pro duct P P of Delaunay p olytop es P and P

is a Delaunay p olytop e and GP P GP GP

Also for every Delaunay p olytop e P there is a pyramid P y r P and if P is

centrally symmetric a bipyramid B P y r P which are Delaunay p olytop es see

DeGr The direct pro duct construction preserves l ness of Delaunay p oly

1

1 1 1

0 0

if GP But H H GP H top es For example GP P

m+m m m

2 2 2

the pyramid and bipyramid constructions can pro duce non gonal Delaunay p oly

top es from those with l skeletons Consider for example the following Delaunay

1

p olytop es

1

n ambo n the Johnson np olytop e P with

n n n n n J

2

GP J n k k bn c for n P r ism P y r

J n1 n1 3 4

3

IR

1

All of them have l skeletons see x for regular ones also ambo H

1 n n+1

2

1 1 1

P H P r ism H P y r H The pyramid and bipyramid construc

J n+1 3 5 4 4

2 2 2

tions pro duce from them l p olytop es in the following cases

1

m m m m

P y r B P y r P y r P y r see x P y r

n n n 4

Additionally we have the following l embeddings

1

1 1 1

2

H P y r ambo H P y r H P y r P r ism

5 n n+1 n1 n1 2n 3

2 2 2

The inclusions i ii b elow show that the skeletons of the mentioned there

p olytop es are non gonal and therefore they are not l graphs

1

2

i K K P y r B P y r P y r ambo n

5 3 n n n

1

2

P y r P since K J n k P y r H n P y r

J 1k n n1 n1

2

1

H n even B P y r ambo n even B P y r

n n

2

ii K P P B P y r P r ism

5 2 3 3 3

Finally the following skeletons are extreme hypermetrics see Prop osition of

1 1

2 2 3 3

DeGr P y r P y r P y r ambo P y r ambo P y r P r ism

5 5 4 4 3

2 2

4

P y r P r ism

3

Small l p olyhedra and examples of p olytopal

hypermetrics

Combinatorial types of dp olytop es were enumerated for small values of v d where

v is the number of vertices see for example x and Tables on p of

Gr u The two prop ositions b elow give l status for p olyhedra of the rst two

1

simple classes

Prop osition i For al l combinatorial types of polyhedra with vertices

we have

a skeletons

G K GP y r rC GB P y r capped K e G

3 4 4 4 3 3 5 3

1

K H

32 4

2

b skeletons

GP r ism K C GP y r rC K P Gcapped K P

3 6 6 5 5 6 5 3 6 4

1

H

5

2

c skeletons K P and K e are non gonal each contains K P P

6 6 32 5 2 3

ii For al l combinatorial types of polyhedra with faces duals of above

we have

a P y r P y r and one with the skeleton K P are selfdual P r ism

3 4 5 6 6 3

B P y r and

3

3 3

1

b P H if GP K P

6 6 4

2

c P is non gonal if GP K P K e Fig

6 5 32

1

H for i One can check that icapp ed dual i capp ed

i+3 3 3

2

1

but dual capp ed truncated is non gonal Also icapp ed H

3 3 3 i+4

2

1

i and dual icapp ed H for i Compare with icapp ed

3 i+6

2

1

H i and for i if no opp osite faces are capp ed it is not gonal

3 6

2

otherwise

Remark that extreme hypermetrics graphs G and G are skeletons of pyramids

1 2

with bases P y r and capp ed see i b ab ove

5 3

Prop osition i For al l combinatorial types of simplicial polyhedra with ver

tices we have

1 1

a capped H capped H

3 5 3 6

2 2

2

b skeletons GB P y r r P are non gonal they contain K P P K

5 5 5 2 3 5

K respectively

3

c skeleton rB of P y r B is the extreme hypermetric G

8 8 4

ii For al l combinatorial types of simple polyhedra with vertices dual of

above we have

1

H if P is a capped capped or B P y r Fig a P

7 3 3 5

2

2

b P is non gonal if GP r P rB Fig

5 8

There are combinatorial types of p olyhedra with vertices having

faces resp ectively The last are simplicial ones The rst two are b oth

non gonal Prop osition ii c

One can enumerate isometric subgraphs and p olytopal ones b etween them of

1

given H which are not isometric subgraphs of a facet of it For example the

n

2

1

following graphs are such isometric p olytopal subgraphs of the Clebsh graph H

5

2

1

C K G together with b elow they are only such subgraphs of H

5 5 4 5

2

on m vertices

K C GP r ism K P p olyhedral see Fig K P

6 6 3 6 5 6 4

Gcapp ed rC GP y r b etween all vertex isometric subgraphs of

3 5 5

1

H

5

2

vertex p olyhedral capp ed P r ism augmented P r ism

3 3

biaugmented P r ism capp ed AP r ism

3 3 4

K capp ed p olytopal P y r P r ism T T K T

2 3 1 3 1

on a facet T

3

2

p olytopal P y r P r ism rT GP y r ambo

3 4

On the other hand all hypermetric but not l graphs with vertices are known

1

see x of DGL they are graphic metrics b etween extreme hypermetrics

Prop osition Between al l the hypermetric graphs on vertices the polytopal

2

ones are only polytopal G rB and polytopal G r C K C G

4 8 1 5 7 5 2

rH K P

2 7 4

Pro of In fact G rH G rB G G G have minimal degree

6 4 8 5 16 24 26

G rB G rH and H K P have minimal degree and are not planar

5 7 7 3 1 6 3

so G rH is not p olytopal also Finally G is planar and has minimal degree

3 1 18

but it can b e disconnected by deleting vertices It is the skeleton of a skew with

a nonplanar face p olytop e 2

Metric of a graph which is hypermetric but not l graph is necessarily an extreme

1

hypermetric The number of vertices of any extreme hypermetric graph is within

Any p olytop e such that its skeleton is extreme hypermetric has dimension

within Call an extreme hypermetric graph of type I of type I I if it generates

the ro ot lattice E E resp ectively A graph G of type I has diameter since

6 7

it is an induced subgraph of the Schlai graph G of diameter and rG is an

21

extreme hypermetric of type I I

k

Clearly GP y r P rGP dimP y r P dimP Remark that P y r C

5

1

H for k f g it is extreme hypermetric for k f g and it is but not

5

2

gonal for k Examples of p olytopal graphs G which are extreme hypermetric

but are not represented as rG for an extreme hypermetric graph G are see x

of type I the p olyhedra M and one with skeleton G see

22 4

Fig

the p olytop es with skeletons G G the p olytop es with skeletons K C

1 2 9 6

1

3 2 2

H the Shlai p olytop e GP y r P r ism r T GP y r ambo r

5 21 3 4

2

having vertices resp ectively

of type I I the p olyhedra M P y r and the p olytop e with

21 4 21

and vertices resp ectively

Some examples of extreme hypermetrics which are not p olytopal but are close

to p olytopal in some sense

a vertex graph G of type I is a planar graph of a skew p olyhedron

18

b The skeleton of the stella o ctangula the section of by with vertices in

3 3

the centers of faces of which is a nonconvex p olyhedron it is vertex graph

3

It contains the induced extreme hypermetric G It is an isometric subgraph of the

x

Goseet graph and since it has diameter it is of type I I

2 3 3

c Antiwebs AW AW are of type I and AW is of type II see xiv

9 12 13

tembeddings in l

A tembedding of a distance kd k is an embedding of the distance kd k kmint d k

ij ij

ij

n

in an l space

1

In what follows describing a tembedding of a p olyhedron P we asso ciate to

every its vertex v a subset av of a set N Usually we take as N the set of all

k gonal faces We say that a face F is reachable by an mpath from a vertex v if

there is an mpath of length m from the vertex v to a vertex of the face F

Truncated icosahedron of diameter has unique DeSp embedding into

1

H asso ciate each vertex to hexagons from all reachable by

20

2

paths resp ectively It is also the unique embedding but not unique embedding

for example asso ciate every vertex to its hexagons

1

H Icosido decahedron of diameter has unique DeSp embedding in

12

2

asso ciate every vertex to p entagons from all reachable by paths

It is also a unique embedding but there is another embedding asso ciate every

vertex to its triangular faces

Cub o ctahedron of diameter has at least two following embeddings

1

in H asso ciate every vertex to its two square faces

6

2

1

in H asso ciate every vertex to its two triangular faces this one as well as two

8

2

1

ab ove ones are tembeddings into H

2D (P )+2

2

Any simple p olyhedron has a embedding in the tetrahedral graph J n so

1

H asso ciate every vertex to its faces If the diameter of a simple p olyhedron in

n

2

is at least then it is embeddable i sizes of its faces are from f g Examples

of this pro cedure are

do decahedron of diameter has a embedding into J also it has a

1

H embedding into

10

2

1 1

J H not unique M J H see Fig g

3 4 8

25

2 2

1

it gives a embedding of P r ism which turns out to b e P r ism H

n n n+2

2

n+2

H for even n

2

Another pro cedure x a wheel rC in the skeleton of an icosahedron and

5

asso ciate every vertex v to the set of all vertices of the wheel at distances and

from v we get a unique embedding of the icosahedron

Another interesting relaxation of our embeddings will b e to consider scaleisometric

embedding into hypercub es of skeletons of simplicial and cubical complexes more

general than the b oundary complexes of p olytop es For example the simplicial

complex on f g with the facets f g f g f g has the skele

ton K K the cubical complex on f g with the facets f g

5 3

f g f g has the skeleton K So b oth are non gonal

33

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Wel AW Wells Structural Inorganic Chemistry Oxford

Zal VAZalgaller Convex polyhedra with regular faces Seminar in Mathematics

of VASteklov Math Institute Leningrad vol Consultants Bureau New

York

APPENDIX

Here we give gures of skeletons of p olyhedra except the p olytop e on Figf

For non hypermetric graphs we indicate on the skeleton p oints violating a k

gonal inequality Actually it is always gonal except Figcf When gures

1

show l p olyhedra the embeddings are into H except the embedding in H on

1 n 5

2

Figa They are shown as follows we lab el a vertex by the sequence i i or

1 m

i i if this vertex is addressed to the set fi i g or fi i g resp ectively

1 m 1 m 1 m

Since m on our gures we use ae as addresses

Figure Non hypermetric p olytop es a M triangular heb esphenoro

20

tunda b M bilunabirotunda c M snub AP r ism d e

8 28 4

dual rhombic icosahedron f Ko esters graph g p olytop e Pyricosahedron PZ

5

Figure Hyp ermetric non l p olyhedra a AP r ism b triaug

1 4

mented P r ism c M spheno corona

3 22 46

1278 78 12

24 5678 1234 68

56 34 3456

28

127 60

6890 3560 347 457 237 157 2347 2478 348 458 7890 3456 3467 1259 4780 236 156 368 568 3456 7890 1278 12 1234 34 126

78

1

Figure a Dual AP r ism H b dual AP r ism c M dual tridi

4 8 4

7

2

minished icosahedron d M dual heb esphenomegacorona e M dual spheno

21 22

1

corona f M dual snub AP r ism gM dual snub disp ensoid H h M dual

4 8

28 25 23

2

1

sphenomegacorona H

10

2

2

Figure Prismatic nonl graphs with minimal n a M b AP r ism c dual

1 8

6

2

AP r ism d AP r ism e dual AP r ism F f T ow g dual

5 6 6 24 3 3

3

2

T ow 3

56 1256 12

57 1268 1234 34

1268 12 57

34 1256 56 1234

1

H and Figure Square ortho and gyrobicup olas C up C up

8 4 4

2

C up C up non gonal

4 4

Figure Dual small non gonal p olyhedra P with GP K e K

32 6

2

P r P rB

5 5 8

234 345 156 157 247 126 467 123 456 137 267 236

347 14578 156 126 346 34578

123 135 456 45678

7

2356 457

23 345 456

12 34 234 156

1256 15 45 3456 237 123 126 167

Figure Dual l p olyhedra dual capp ed dual capp ed

1 3 3

1 1 1 1

H H H H resp

7 8 6 7

2 2 2 2

Figure All l skeletons of basic regularfaced p olyhedra M i

1 i

Figure All not hypermetric skeletons of basic regularfaced p olyhedra M l i

i

126789 1289 12890a

6789 12 1234 67 12340a

35 67de 1345 12340abc

1345bc 3567de 35de 13450abc

1345de 1345bcde

12579a 46

680b 579a 60 460a 259b 59 12579b 127 469a 23580b 250b 9a 0b 157 126 45 1 159a 5 6 23680b 360b 236 5b 159b 360a 6a 457 456 346 15 580b 34670a 679a 26 12

34679a

2 3

Figure Fullerens a T ow b dual T ow c strained F D d F T

30 5h 28 d

5 5

1 1 1

e dual F f b owl H g F H h dual F T H

26 15 26 12 28 d 7

2 2 2 2 23 13 12 3 1 45 15 14 3 5 4 24 1 35 25 34 2 34 25 35 24 1 4 5 4 9 3 2 15 45 13 5 8 12 6 7 23

14

Figure a ElDo elongated do decahedron H b the graph on H n ElDo c

5 5

2 2

AW antiweb AW a hypermetric nonl graph

1

9 9