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