A zoo of l1-embeddable polytopal graphs - CiteSeerX

A zoo of l1-embeddable polytopal graphs M.Deza CNRS, Ecole Normale Superieure, Paris V.P.Grishukhin CEMI, Russian Academy of Sciences , Moscow Abstract

A simple graph G = (V; E ) is called l1 -graph if, for some ,n 2 N, there exists a vertex-addressing of each vertex v of G by a vertex a(v) of the n-cube Hn preserving, up to the scale , the graph distance, i.e. dG (v; v0 ) = dHn (a(v); a(v0 )) for all v 2 V . We distinguish l1 -graphs between 1-skeletons of a variety of well known classes of polytopes: semiregular, regular-faced, zonotopes, Delaunay polytopes of dimension  4 and several generalizations of prisms and antiprisms.

1. Introduction 2. Some notations and properties of polytopal graphs and hypermetrics 2.1 Vector representations of l -metrics and hypermetrics 3. Regular-faced polytopes 3.1 Regular polytopes 3.2 Semiregular (not regular) polytopes 3.3 Regular-faced (not semiregular)polytopes of dimension  4 4. Prismatic graphs 4.1 Moscow and Globe graphs 4.2 Stellated k-gons 4.3 Cupolas 4.4 Antiwebs 4.5 Capped antiprisms, towers and fullerenes 5. 92 regular-faced (not semiregular) polyhedra 6. Zonotopes 7. Delaunay polytopes 8. Small l -polyhedra and examples of polytopal hypermetrics 9. t-embedding in l Appendix: gures 1

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The authors has been supported by DFG{Sonderforschungsberich 343 \DISKRETE STRUKTUREN IN DER MATHEMATIK" UNIVERSITA T Bielefeld as well as by DIMANET{THE EUROPEAN NETWORK ON DISCRETE MATHEMATICS, EU Contract #ERBCHRXCT 940429 

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1 Introduction A connected simple graph G is called an l -graph if its (shortest path) distance matrix dG, seen as a metric space (d; X ) on the set X of vertices of G, is embeddable isometrically into some lm -space. The lm-space is Rm with l -metric 1

1

1

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dl1 (x; y) = jjx ? yjjl1 =

m X i=1

jxi ? yij:

Equivalently, see [AsDe80], G is an l -graph i dG is embedded isometrically into the restriction of ln on an n-cube up to a certain scale  . Clearly,  = 1 (respectively,  = 2) means that G is an isometric subgraph of the n-cube graph (respectively, of the n-half-cube graph). We denote these embeddings by G ! Hn (respectively, G ! Hn). Any metric subspace A of lm is (see [Dez60]) hypermetric, i.e. it satis es 1

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1 2

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X

i (d(x; a)+ d(x; b)+ d(x; c))+(d(y; a)+ d(y; b)+ d(y; c)). They are [0,2;11,12,14] for m = 2, [0,3;11,22,24] for m = 3, [0; m; 11; dm=2e3; (m ? 1)5] for m  4. (i) above take the form: for m = 2: 2 + (1 + 2 + 2) > 3 + 3, for m = 3: 3 + (2 + 3 + 2) > 5 + 4, for m  4: m +((1+ dm=2e)+(m ? 2+1)+(m +1 ?dm=2e) = 3m +1 > (1+ dm=2e +(m ? 1))+((m ? 1) + (m ?dm=2e) + 1) = 3m if n = 5, m + ((1 + dm=2e) + (m ? 2 + 2) + (m +1 ?dm=2e) = 3m + 2 > (1 + dm=2e + (m ? 1)) + ((m ? 1) + (m ? dm=2e) + 1) = 3m if n  6. 2 11

Finally, we consider the graph 1-Mnm :=2-Mnm with the deleted vertex m, i.e.1-capped Moscow graph. It is the usual pyramid for m = 1. Clearly, it is the skeleton of a self-dual polyhedron. (

n? if n = 3; 4; 5 Corollary 1 1-Mnm !is Hnotm 5-gonal for n  6 Proof. In fact, 1-Mnm is an isometric subgraph of 2-Mnm if n  5. But, the cycle 1 2

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+

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C m? ;:::; m? n is not an isometric subgraph (the polar way via 0 is shorter than within this cycle) i bn=2c > 2(m ? 1). So, embeddings for n = 3; 4 come by deleting the vertex m. For n = 5, an embedding is given by a(0) = ;, a(ij ) = fj; j + 1((mod 5)g [ ft 2 Z : 6  t  2i + 3g, for 1  i  m, 1  j  5. For n  6, one can nd (see Fig.4a for n = 6) 5 points violating 5-gonal inequality. 2 Another (than Moscow graph) generalization of Prismn = C  Cn is the Lee graph Qt i Cni ; its pathPmetric is the Lee distance dLee (x; y) = ti min(jxi ? yij; ni ? jxi ? yij). used in the coding theory (a discrete analog of elliptic metric). Remind that H = C  C , since C = HQ . t C ! H (and moreover, ! H n if all n are even), where n = Pt n . Clearly, n ni i i i i 2 Qt Pt Pni ! Hn?t ; it is a grid graph with usual grid distance, i.e. l -distance i jxi ? yij. Qit i Kni ! Hn?t ; it is the Hamming graph with Hamming metric jf1  i  t : xi 6= yigj. (

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4.2 Stellated k-gons

Stellated k-gon Stelk is the 2k-cycle C1;10 ;2;20 ;:::;k;k0 with the additional edges of the kcycle C1;2;:::;k. So, Stelk is Ck with a triangle on each edge. Besides Stelk is APrismk without the edges of the k-cycle C10;20;:::;k0 . Also it is an isometric subgraph of skeletons of many important polyhedra, for example, of icosahedron and 3 others (with 8,9,10 vertices) convex deltahedrons for k = 3, of cuboctahedron and snub cube for k = 4 of icosidodecahedron and snub dodecahedron for k = 5, of Archimedean tiling (3,6,3,6) for k = 6. For n = 5; 6; 8 it is used as well-known symbol (Red Star, Star of David, Muslim Star, respectively). Denote by Sun2k the isometric subgraph of Stel2k obtained by deleting vertices (2i?1)0, 1  i  k. It is an isometric subgraph, for example, of truncated tetrahedron, truncated cube, of Archimedean tiling (3; 122) for k = 3; 4; 6, respectively.

Proposition 2 (i) Stelk ! H k , (ii) Sun k ! H k . Proof. In fact, let i; i0, 1  i  k, denote vertices. For odd k = 2m + 1, an embedding is given by a(i) = fi; i + 1; :::; i + m ? 1g, a(i0 ) = fi; i + 1; :::; i + m; 2m + 1 + ig, for 1  i  k 2

1 2

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(addition is modulo k, except for 2m + i + 1). For even k = 2m we de ne a(i) inductively. We take a(m + 1) = f1; :::; kg. Then a(i) = a(i + 1) with two largest elements deleted if 1  i  m, a(i) = a(i ? 1) with two smallest elements deleted if m + 2  i  k. 12

a(m0 ) = f1; :::; k ? 1; k + mg, a((m + 1)0) = f2; :::; k; k + m + 1g, a(i0 ) = fk + ig[ (a((i +1)0) ?fk + i +1g) with two largest elements deleted if 1  i < m, a(i0 ) = fk + ig [ (a((i + 1)0) ? fk + i ? 1g) with two smallest elements deleted if m + 1  i  k,

4.3 Cupolas

Denote by Cupn the graph with vertices i, i0 , i00 (1  i  n) and edges on the inner cycle C ;:::;n, on the outer cycles C 0 ; 00;:::;n0;n00 , and on the paths Pi0 ;i;i00 for all 1  i  n. So Cupn is the skeleton of a polyhedron; for n = 3; 4; 5 they are triangular, square and pentagonal cupolas: the polyhedra ##23,24,25 in the list of 112 regular faced polyhedra. 1

1 1

Proposition 3 (i) Cupn ! H n for n  4, (ii) Cup and Cupn, n  3, are not 5-gonal. Proof. In fact, Cup contains a not 5-gonal isometric subgraph v(3; 4; 3; 4) (see de nitions in x5) on 7 points. We give the following explicit embedding of Cupn ! H n. We take an embedding of the inner cycle C ;:::;n ! Hn. Let the vertex i is mapped in this embedding into an subset a(i) of an n-set V . Let a(1) = ;, and V \ f1; 2; :::; ng = ;. Then we map the vertices i0 and i00 into the sets a(i) [ fi; i + 1g and a(i) [ fi + 1; i + 2g, respectively (the addition 1 2

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is by mod n). Note that the shortest path between non-adjacent vertices of the outer cycle goes through the inner cycle. Hence, it is not dicult to verify that the obtained embedding is isometric for n  4. Cupn contains K ; as an isometric subgraph for n  3. 2 23

4.4 Antiwebs

An antiweb AWnk , for 1  k  b n c, is the graph with the vertices 0; 1; :::; n ? 1 and i  i + 1; i + 2; :::; i + k (mod n). (Here i  j denotes that the vertex i is adjacent to the vertex j .) It is a common generalization of the following polyhedral graphs: AWn = Cn ! Hn (and ! H n2 if n is even), AWn = G(APrism n2 ) ! Hn if n is even, AWnbnn= c = Kn = G(n? ) ! Hn, AWn2 ? = K n2  = G( n2 ) ! m H m (for some m) if n is even. In the next proposition we show that all other antiwebs are not l -graphs. It is easy to check that AWnk has diameter 2 i b n c  k  b n c ? 1. 2

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Proposition 4 (i) AWn is not 5-gonal for odd n  13, AW is not 7-gonal. 2

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(ii) AWnk is not 5-gonal in the following cases: e  k < n2 ? 1, (a) if d n+1 4 (b) if n = 4k + 3 for k  3, and if n = 4k + 4 for k  4. (iii) AWnk is not l1 -embeddable if either k = 3 and n  10 or 4  k  n?2 3 (and therefore n  11). In particular, AW133 and AW164 are not 7-gonal. (iv) AW92 , AW123 and AW143 are extreme hypermetrics.

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Proof. (i) We show explicitly 5 points (a; b; c; x; y) of AWn , where the 5-gonal inequality (1) with ba = bb = bc = ?bx = ?by = 1 is violated. If n = 4t ? 1  15, take the points (0; 2t ? 1; 2t; 2t ? 4; 2t + 3). If n = 4t + 1  13, take the points (0; 2t ? 1; 2t + 2; 2t ? 2; 2t + 3). 2

Similarly, for AW we have the following 7 points (0,5,6,11;3,4,9) violating a 7-gonal inequality. (ii) a) In fact, any Akn with k as in Proposition 4 is of diameter 2, so any its induced subgraph of diameter 2 is isometric. Only connected graphs on 5 vertices which are not 5-gonal are K ; , K ? K and K ? P ? P (with P \ P = ;). Hence the only way for AWnk of diameter 2 to be not 5-gonal is to contain one of these 3 graphs on 5 vertices. But any antiweb is K ; -free graph. Hence the rst 2 above subgraphs, K ; and K ? K , are excluded. On the other hand, we will give subgraph K ? P ? P , namely, the complement of P ; k?i; k? i? + Pk?i;n ?k, for any i with n +1+ i  4k  n +1+2i and 0  i  k ? 2. So, it will cover all cases n + 1  4k  n ? 3 + 2k, i.e. d n e  k  b n? c considered in Proposition 4. Now, P ; k?i; k? i?  AWnk , since 1 + k < 2k ? i, (2k ? i) + k < 4k ? 2i ? 1 (because i  k + 2) and 4k ? 2i ? 1  n, (4k ? 2i ? 1) + k ? n  1. Pk?i;n ?k  AWnk , since 1  k ? i, (k ? i) + k < n + 1 ? k  n. But k ? i; n + 1 ? k  1; 2k ? i; 4k ? 2i ? 1 in AWnk . (ii) b) If n = 4k +3, take the 5 points (0; 2k +1; 2k +2; k +2; 3k +3), and if n = 4k +4, take the 5 points (0; 2k + 1; 2k + 2; k + 2; 3k + 4). (iii) For k = 3, consider the subgraph of AWn induced by the points i, 0  i  6. Here the point 3 is adjacent to all other 6 points. Since n  10, the point 6 is not adjacent to the points 0,1,2. Finally we have that the induced subgraph is rB . For k  4, consider the subgraph of AWnk induced by the following 7 points: 2; 3; 4; k; k + 2; k + 3; k + 4. Since k  n? , this graph is r(K ? P ) = rH , where P = P ; ; ; . The graphs B and H are the graphs on Figs. 6.3 and 6.7 of [DGL95]. Since rB = G and rH = G are graphs from the list of the 26 extreme hypermetrics on 7 points, we have that AWnk is not l -embeddable. The following 7 points violate a 7-gonal inequality: (0,4,5,6;2,3,7) for AW and (0,2,5,7;3,4,9) for AW . (iv) The graph AW has 3 hexagons (0,2,3,5,6,8), (0,1,3,4,6,7) and (1,2,4,5,7,8) determining 3 6-gonal equalities. Any two from them are linearly independent. Hence representations of two vertices, say, 7 and 8, are determined by representations of other vertices. If we delete the vertices 7 and 8, we obtain the extreme hypermetric G from Fig.5.14 of [DGL95]. Since AW has diameter 2, it is a subgraph of the Shla i graph. For example, the following labelings the vertices 0,1,...,8 by the pairs 13, 12, 14, 24, 45, 17, 56, 36, 48 gives an embedding. A propos, AW ? e is a polyhedral graph (see Fig.11c). Similarly, we nd that the vertices i, 0  i  6, of the graph AW induce the extreme hypermetric graph G = rB . The following labeling shows that AW is an extreme hypermetric graph: 0 ! 12, 1 ! 13, 2 ! 14, 3 ! 15, 4 ! 26, 5 ! 45, 6 ! 57, and 7 ! 16, 8 ! 18, 9 ! 27, 10 ! 23, 11 ! 58. The graph AW has diameter 3 and the set of its vertices consists of 7 pairs of vertices at distance 3. The 7 vertices i, 0  i  6, induce the extreme hypermetric graphs 2 11

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G = rB . Hence AW is not l -graph. But the following labeling  shows that it is extreme hypermetric: 0 ! 12, 1 ! 13, 2 ! 14, 3 ! 15, 4 ! 26, 5 ! 23, 6 ! 57, and i + 7 ! (i). 2 4

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4.5 Capped antiprisms, towers and fullerenes

We consider here an antiprism analog of capped prisms (1-Mn , 2-Mn of x4.1). Take at rst APrism = . The i-capped ! Hi , (0  i  8). In fact, they are isometric subgraphs of omni(8)-capped (= dual truncated ). Denote the vertices of by i; i0, where 1  i  3 and all (i; i0 ) are not edges, and the caps of faces (1; 20; 30), (10; 2; 30), (10; 20; 3), (1,2,3) by t and the caps of respectively opposite faces by t0, 1  t  4. Put then all a(i) = fi; 4g, a(i0 ) = f1; 2; 3; 4g ? a(i), a(t) = ft; t + 4g, a(t0 ) = (f1; 2; 3; 4g ? ftg) [ f8 + tg. The dual i-capped ! Hi for 0  i  2, but it is not 5-gonal for i = 8. >From now on we consider i-capped APrismn, n  4, only for i = 1; 2 (i.e. only n-cycles are capped). Denote by i-APrismn i-capped APrismn for i = 1; 2. Remind (x3.2) that APrismn ! Hn for n  3 and dual APrismn is not 5-gonal if n  4. The next case is APrism = AW ; it can be seen as a half of the Shrikhande graph. i-APrism is an extreme hypermetric for i = 1; 2 (see x8 and Fig.2); they are ##30,37 of the list of 92 regular-faced polyhedra. But their dual are scale 2 embeddable into 8-, 9-cube, respectively (see Fig.3). i-APrism ! H for 0  i  2. Its dual (dodecahedron)! H for i = 2, but it is not 5-gonal for i = 0; 1 (Fig. 4c). 1-APrismn ! Hn for n = 5; 6; 7, but not 5-gonal for n  8. In fact, remind that vertices of APrismn are i; i0 for 1  i; i0  n and edges are pairs (i; i0 ), (i + 1; i0). Let 0 be the cap of C ;:::;n. Put then a(0) = ;, a(i) = fi; i + 1g, a(i0 ) = fi; i + 1; i + 2; n + 1g. Clearly, for n = 5; 6; 7 it will give a scale 2 isometric embedding into (n + 1)-cube. For n  6 dual 1-APrismn, 2-APrismn and their duals are not 5-gonal (see Fig.4d,e for case n = 6). Consider now towers, generalizing gyroelongations of antiprisms in a way similar to how the Moscow graph in x4.1 generalizes elongations of prisms. Denote by Townk the polyhedron de ned by adjoining consecutively k copies of APrismn one on the top of another. It is easy to check that Town is not 5-gonal for n  3: see Fig. 4f,g for Tow (i.e. gyroelongated , it appears also on Fig.16.1.1 in [Gru67]) and its dual. It implies that Townk and 2-capped Towk are not 5-gonal for k  2. Remark similarity of 2-Towk with polar zonotopes PZm mentioned in x6. See also Fig.3e for non 5-gonal Koester's graph, [Koe85], similar to APrism + APrism and to PZ ; it is 4-regular planar 4-critical graph, i.e. its chromatic number is 4, but every its proper subgraph has a 3-coloring. Dual 2-Towk is a fullerene, i.e. a simple polyhedron with 20 + 2m vertices, 12 pentagonal and m hexagonal faces) F m for m = 5(k + 1). Let us call it strained (most \antiaromatic", least stable, in chemical terms), since it has (as opposite to preferable fullerenes) maximum number of abutting pairs of pentagonal faces. Except the case k = 1 (dodecahedron) the strained fullerene F k is not 5-gonal: for k  3 it contains the graph in Fig. 10b, for the case k = 2 see Fig. 10c. Its dual also not 5-gonal since it 2

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contains 1-Tow depicted on Fig. 10a. We give below some observations on l -status of several fullerenes. In particular, small fullerenes F , F , F are of special interest for chemistry: besides being carbon cages, they are used ([Wel84], p.74 and pp.660{662 on \ice-like" clathrate hydrates) as space co- lling polyhedra. There are [Bal95] 1, 0, 1, 1, 2, 3, 40, 271, 1812 fullerenes F m for m = 0, 1, 2, 3, 4, 5, 10, 15, 20, respectively. F is the dodecahedron; the unique F is the dual of 2-APrism (it and F  are on Fig. 4e,d); the unique F and its dual are on Fig. 10g,e; F (Td) (Td is its group of symmetry) and its dual are on Fig. 10d,h; F (D h) is on Fig. 10c. The strained fullerenes F m for m  25 are F (D h), F (D d ), F (D h), F (D d), F (D h); with the symmetry it is unique for m = 5; 20 and one of two for m = 10; 15; 25, respectively. The truncated icosahedron with its icosahedral symmetry is de ned as F (Ih). As we checked, all above fullerenes and their duals are not 5-gonal, except F ! H , F ! H , F  ! H , F  (Td ) ! H , F  (Ih) ! H . F (Ih) admits 7-embeddings in H (see [DeSp96] and x8 here). Using also the algorithm of [DeSp96], one can see that the unique preferable (i.e. without abutting pairs of pentagons) F is not l -graph. This F is second one with the symmetry D h and contains 20-bowl, i.e. the graph consisting of a pentagon surrounded by 5 hexagons; 20-bowl ! H (see Fig.10f). Also the graph consisting of a pentagon surrounded by 5 pentagons (or by 3 pentagons and 2 non-adjacent hexagons) is embeddable into ! H , (! H , respectively). 2 5

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5 92 regular-faced (not semi-regular) polyhedra We use names and numbers of those polyhedra from [Ber71], where they are given as # #21{112 of the list of all 112 regular-faced polyhedra; in [Zal69] they numbered by 1{92. In both references they are given as same appropriate joins of 28 basic polyhedra M ?M . See also [KoSu92] with nice pictures of those polyhedra. 28 basic regular-faced polyhedra (all but M are given on Fig.8,9) include regular ones M , M ; pyramids M , M , cupolas M ? M , The 9 Archimedean polyhedra Mi denoted as 1; 3; 30; 2; 5; 50; 20; 7; 70 in Table 1 of x3.2 for i 2 f10 ? 12; 16 ? 19; 26; 27g, respectively; M , M which are tridiminished, parabidiminished RIDo, M which is tridiminished Ico; pentagonal rotunda M and 8 others whose complicate names re ect their high individuality. In this section we denote icosahedron, dodecahedron and rhombicosidodecahedron by Ico, Do and RIDo, respectively. Remark that remaining 4 Archimedean polyhedra of Table 1 are in this notations 4=Cup + Cup , 40 = M + M , 60 = RIDo = M + 2Cup , 6 = Cup + Prism + Cup . 1

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Lemma 3 (i) Mn is not 5-gonal for n = 4; 8; 10 ? 13; 19 ? 21; 23 ? 25,

(ii) M28 (= #105, snub square antiprism) is not 7-gonal, (iii) M22 (= #106, sphenocorona) is extreme hypermetric, (iv) Remaining 15 polyhedra Mn have l1 -skeletons. (v) Mn has l1 -skeleton for n = 1 ? 3, 10{12, 15, 16, 19, 23, 25 and it is not 5-gonal for remaining 17 polyhedra Mn .

Proof. (i),(ii) are clear from Fig.9; (iii) is proved in the proof of the next lemma. 14 l polyhedra are M , M (see x3.1); M ? M and M , M (see x3.2); M ! H , 1

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M ! H and M ! H , M ! H (see xx4.1,4.3); the skeleton of M is an isometric subgraph of l -skeleton of Ico, M ! H . Finally, M = #26 ! H , where M  #41 ! H . 2 1 2

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Lemma 4 Skeletons of the following 4 polyhedra are extreme hypermetrics:

#30=APrism4 + Pyr4 (gyroelongated Pyr4= 1-capped APrism4 ), #37=Pyr4 + APrism4 + Pyr4 (gyroelongated BPyr4= 2-capped APrism4 ), #71=APrism3 + 3Pyr4 (triaugmented Prism3 ), #106=M22 (sphenocorona), #107=M22 + Pyr4 (augmented sphenocorona).

Proof. The skeletons of the polyhedra #30 and #71 have each 9 vertices and diameter 2. The skeleton of #37 has 10 vertices, diameter 3 and contains the skeleton of #30 as an isometric subgraph. These 3 graphs contain the induced extreme hypermetrics G and G of [DGL95] Fig.5.15. This implies that these skeletons are not l -graphs. We give explicit embeddings of skeletons of #30 and #71 into the Schla i graph G(2 ), and the skeleton of #37 into the Gosset graph G(3 ); it will show that these skeletons are extreme hypermetrics. Recall that the vertices of APrism are i; i0, 1  i; i0  4, with the edges (i; i + 1), 0 (i ; (i + 1)0), (i; i0 ) and (i + 1; i0) (here and below additions are mod 4). Let 0 be the new vertex of the cap of the capped antiprism #30, and 0' be the second cap of the 2-capped antiprism #37. Then the labels of vertices by pairs are as follows: a(i) = i; i+1, 1  i  4, a(10) = 15, a(20 ) = 47, a(30 ) = 36, a(40) = 28, a(0) = 24, a(00) = 24. Similarly, let i; i0 , 1  i; i0  3 be the vertices of the prism of #71 with edges (i:i0), and let i00 be the apex of the Pyr on the square face of Prism opposite to the edge (i:i0 ). Then the labels are: a(i) = i7, a(i0 ) = i8, a(i00) = (i; i + 3), 1  i  3. The skeleton of #107 has 10 vertices and diameter 3; it contains the induced extreme hypermetric G . It is 1-capped APrism with an additional edge between two nonadjacent vertices of the noncapped cycle C . Let, as above, the vertices of APrism are i; i0, 1  i  5, such that all vertices i are adjacent to the apex 0 of the cap, and the vertices 10 and 40 are adjacent. We label the vertices by pairs as follows: a(i) = i; i + 1, 1  i  5. Since the vertices 20, 30 and 50 are at distance 3 from the vertices 4, 5 and 2, respectively, we have a(20) = 45, a(30 ) = 15 and a(50) = 23. The other two adjacent vertices obtain the labels a(10) = 16 and a(40) = 46. The skeleton of #106 is G(#107) without the vertex 50, which is the apex of the cap Pyr . The labeling of G(#106) is induced by the above labeling of G(#107), since G(#106) is an isometric subgraph of G(#107). 2 24

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Proposition 5 Besides non 7-gonal #105 and above 5 hypermetrics ##30, 37, 71, 106,

107, remaining 86 polyhedra consist of 50 not 5-gonal ones and 36 with l1 -skeletons: ##21{35 (except 23,30,33), 39, 40, 41, 44, 48, 69{84 (except 71, 77), 92{95, 100.

Corollary 2 l -status of 8 convex deltahedra is as follows: (i)  ! H , #32=BPyr ! H , ! H , Icosahedron ! H , 3

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(ii) #33=BPyr5 and #104=M25 (snub dispensoid) are not 5-gonal, (iii) #37, #71 are extreme hypermetrics.

17

Remark that all 5 dual truncated P , where P is a Platonic solid, are simplicial and have l -skeletons. They ! Hn with n = 7; 6; 12; 10; 16 for P being  , , , Ico, Do, respectively. Between other simplicial Catalan polyhedra, namely Prismn and dual truncated Q (Q being cuboctahedron or icosidodecahedron), only Prismn (n = 3; 4) are 5-gonal; between remaining 9 Catalan polyhedra (including APrismn, n  4) only dual Q are 5-gonal. Sketch of the proof of Proposition 5. a) 12 polyhedra ##92{103 They are all possible adjoinings of Cup = M to 10-gonal faces of two of them: M = #103 and M = #100 (see Fig. 8). Also Archimedean RIDo = M + 2M = M + 3M ! H . One can check that slight modi cations of this embedding produce an embedding into the same H of M and all 4 polyhedra without 10-faces: ##92{ 95=M + M + M , M +2M , M + M +2M , M +3M . The 7 remaining polyhedra are not 5-gonal: see Fig.8 for M . Slight modi cations of these 5 points give, due to symmetries, forbidden 5-points con gurations for others. b) not 5-gonality For 22 polyhedra (# # 43, 45, 49{54, 57{63, 66, 77, 108{112) 5 vertices violating 5-gonality were found ad hoc. ##108{112=Mi for i = 23; 21; 24; 8; 20 (see Fig.9); see Fig.5b for #49. Denote by v(3; 2k; 3; 2m) (respectively, by e(3; 2k; 3; 2m)) the graph consisting of a vertex (respectively, of an edge) surrounded by cycles C , C k , C , C m (for positive integers k, m) in this order; each two consecutive cycles intersect exactly in the edge incident (adjacent) to the original vertex (respectively, to the original edge). It is easy to check that both the families of the above graphs are not 5-gonal. Now v(3; 4; 3; 4)  #23 = M  ##38; 42; 47(\anticuboctahedron00 ); 55; 56; 64; 85; v(3; 4; 3; 4)  #46; e(3; 8; 3; 8)  #10 = M  ##86; 87; e(3; 10; 3; 10)  #12 = M  ##88 ? 91; #43  #65; #45  ##67; 68; K ? P ? P  ##33; 104 = M ; #36 = 2-M (so, not 5-gonal, see x4.1). #77 is easy to check. Remaining 15 polyhedra have many vertices; they are too cumbersome for to present them on gures. So we leave them to the reader and give here only a helpful clari cation of them in terms given in Remark 1 below. ##50, 54 are orthobi-Cup , -M ; #51 is gyrobi-Cup ; ##58, 62 are elongated orthobi-Cup , -M ; ##57, 59, 63 are elongated gyrobi-Cup , -Cup , -M ; ##43, 45 are gyroelongated Cup , M ; #66 is gyroelongated bi-Cup ; #52=Cup + M , #53=Cup + M and #60=Cup + P + M , #61=Cup + P + M . c) l -embeddings For 9 polyhedra (##39{41, 44, 48, 70, 73, 75, 76) l -embeddings were found ad hoc. Now #24 = M = Cup  #39; #25 = M = Cup  #40; #26 = M  #41; #69  #70; #72  #73; #74  #75; ##31; 82; 83  Ico; #84  dual truncated Do ! H ; #84 ! H : 1 2

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##78 ? 81  dual truncated Ico ! H . More precisely, ) ##31; 82 ! H , since APrism  #31 = APrism + Pyr  Pyr + APrism + Pyr = Ico; #83 = M  #82 = M + Pyr  M + 3Pyr = Ico and M ; APrism ; Ico ! H ; ) ##78 ? 81 ! H , since Do, dual truncated Ico ! H , Do  #78 = Do + Pyr  #79 = Pyr + Do + Pyr  dual truncated Ico, #78  #80 = Do + 2Pyr  #81 = Do + 3Pyr  dual truncated Ico. We have #21 = M = Pyr ! H ; #22 = M = Pyr ! H ; #32 = BPyr ! H ; #28 = 1-M ! H ; #29 = 1-M ! H ; #27 = 1-M ! H ; #34 = 2-M ! H ; #35 = 2-M ! H : 1 2

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(See x4.1 for those embeddings.) Now we present 9 embeddings ad hoc. Between them: #39 = Cup + Prism ! H ; #40 = Cup + Prism ! H ; #44 = Cup + APrism ! H ; #48 = 2Cup ! H ; #41 = M + Prism ! H : 4

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An embedding #48 ! H is given on Fig.5a. 4 others are obtained by capping: #70=biaugmented Prism (i.e. capped on 2 square faces)! H , #73=biaugmented Prism (on two non-adjacent square faces)! H ; ##75,76=parabiaugmented, metabiaugmented Prism (on 2 opposite or non-opposite non-adjacent square faces) ! H . (If we relax de nitions of ##73, 74, 76, 77 by permitting to cap adjacent square faces, then we get non-convex polyhedra having same l -status of the skeletons as original ones.) Elongated polyhedra ##39, 40, 41 are easy (see Remark 1). We leave #44 and above 4 capped polyhedra to the reader. 2 Remark 1. For a polyhedron P (following, for example, pp.349{351 of [Ber71]) call P + P , P + P , P + Prism, P + APrism, P + Prism + P , P + Prism + P , P + APrism + P , respectively: orthobi-P , gyrobi-P , elongated-P , gyroelongated-P , elongated orthobi-P , elongated gyrobi-P , gyroelongated bi-P . Here ortho (gyro, respectively) means that two solids are joined together such that one of two bases is the orthogonal projection of (is turned relative to, respectively) the other. Prisms and antiprisms above have an appropriate base and are adjoined by it. List of 92 regular-faced polyhedra contains orthobi-P and elongated orthobi-P for exactly P = Mi (i = 1 ? 6; 9); it turns out that both have l -skeletons if i = 1; 2; 5 and both are not 5-gonal otherwise. All gyrobi-P , gyroelongated bi-P and elongated gyrobi-P in the list are not 5-gonal. Between gyroelongated polyhedra of the list, two (#31=Pyr + APrism and #44=Cap + APrism ) have l -skeletons and two (#30=Pyr + APrism and #37=Pyr + APrism + Pyr ) are extreme hypermetrics. Clearly, elongated P has l -skeleton i P has. We have (see Fig.5) that #49 = Cup + Cup is not 5-gonal, while #48 = 2Cup ! H and elongated #49 = Cup + Prism + Cup ! H (see Remark 1 in x3.2). Remark 2: duals of regular-faced polyhedra. 1 2

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Examples of non 5-gonal ones are duals of ##23-25 (cupolas M ? M ), #46 (gyrobifastigium); ##26, 83, 100, 103, 105, 109{112= Mi , (i = 9, 7, 14, 13, 28, 21, 24, 8, 20). M  is even strongly not 5-gonal: its skeleton, as well as K ; , contains 5 vertices x; y; a; b; c with each of x; y being on the same shortest path between vertices of the pairs (a; b), (a; c), (b; c); both M  and K ; have only square faces. M  has only (16) pentagonal faces. Other above Mi have more than one type of faces; for example, M  , dual disphenocingulum, has only square and pentagonal faces. Examples of l -graphs are duals of l -graphs ##21{22, 27{29, 32{35 and non 5-gonal ##36, 104 = M ; 108 = M . M  ! H , M  ! H ; see Fig. 3g,h, M  ([Wel84], p.75) together with a 17-hedron lls R . Duals of ##34{36 are Mn for n = 3; 4; 5; see x4.1. Also, the duals of the extreme hypermetrics ##30; 37 ! H ; H , respectively (see Fig.3), while duals of ##71, 106=M , 107 are not 5-gonal. It is interesting to characterize pairs (P; P ) of dual polytopes having both l -skeletons. Examples: (n; n), ( n; n), (1-Mnm ,1-Mnm ) for n 2 f3; 4; 5g, (icosahedron, dodecahedron), (truncated octahedron, its dual), (i-capped  , its dual) for i 6= 4, (i-capped , its dual) for i  2, (Pm  Cn, 2-Mnm? ) for n 2 f3; 4g (notations are from x4.1). Remark 3. Besides fullerenes F m (see x4.5) majority of chemically relevant polyhedra (i.e. most frequent ones as arrangement of nearest neighbours in crystals, molecules or ions) are regular-faced. They include 8 deltahedra (see Fig.1 in [SHDC95] and Corollary 2 above), Prism (also its dual and its augmentations ##69{71),  , Pyr , cuboctahedron, #104 and capped (see x4.5) antiprisms (i-capped APrismn for n = 3; 4 and 0  i  2; 2-capped APrism = icosahedron; 2-capped APrism , dual of the fulleren F , see x4.5. Regular-faced polyhedra are also used in large chemical literature on metal clusters; see, for example, [DDMP93] considering ##25, 35, 37, 47, 49 (see it on Fig. 5b), 55, 56, 58. 4

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6 Zonotopes

Let ei, 1  i  n, be n mutually orthogonal vectors of a same length. Let e(S ) = Pi2S ei for S  N = f1; 2; :::; ng. Then the convex hull of the vectors e(S ) for all S  N is the n-cube n. The 2n points e(S ) are vertices of n. If we project n onto a k-dimensional space for k  n, we obtain a zonotope Zn. The zonotope Zn has the following three equivalent characterizations: it is 1) a projection of an n-cube, 2) a Minkowski sum of line segments, 3) a polytope having only centrally symmetric faces. Let vi be the projection of ei. Without loss of generality we may assume that vi 6= 0 for all i P2 N . Denote the zonotope Zn also by Z (v ; v ; :::; vn) = Z (vi : i 2 N ) The point v(S ) = i2S vi is the projection of the vertex e(S ). Let all the vectors vi go out from an origin. Then the origin is the point v(;). We can take any point v(S ) as a new origin. This is equivalent to a change of signs of the vectors vi for i 2 S . With the origin in v(S ) the zonotope Zn takes the form Z (?vi : i 2 S ; vi : i 2 N ? S ), and the point v(T ) is transformed into the point v(S
T ). The point v(S ) is a vertex of Zn not for every S . There is the following simple criterion when a set S determines a vertex of Zn. Since every vertex is an extreme point, for every 1

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vertex v(S ) of Zn there is a k-vector c such that cv(S ) > cx for all x 2 Zn, x 6= v(S ). (Here cx is the inner product of vectors c and x). In particular, this implies that cvi > 0 for all i 2 S , and cvi < 0 for all i 2 N ? S . In other words, the k-dimensional hyperplane fx : cx = 0g supporting the vertex v(S ) separates all vectors vi for i 2 S from all other vectors. It is easy to see that two vectors v and v0 are separated by a hyperplane if and only if they have distinct directions, i.e. v0 6= v for some real   0. If vk and vj have a same direction, then Z (vi : i 2 N ) = Z (vk0 ; vi : i 2 N ? fj; kg), where vk0 = vk + vj . Hence without loss of generality we may suppose that all vectors vi have distinct directions. We denote the family of all subsets S determining the vertices of Zn by Sn . If the origin is the vertex v(;), then all vectors lie in a halfspace determined by the hyperplane supporting v(;). Without loss of generality, we can suppose that the origin is a vertex of Zn, i.e. ; 2 Sn . Let V  N be such that v(fig) = vi is a vertex for all i 2 V . Then Zn lies in the conic hull of vi for i 2 V .

Lemma 5 S \ V 6= ;, for every S 2 Sn. Proof. Let S 2 Sn, and let c be a vector such that cvk > 0 for k 2 SP, and cvi < 0 for i 2 N ? S . Every vk lies in the cone generated by vi , i 2 V , i.e. vk = i2V i(k)vi with i(k)  0. If S \ V = ;, then the inequality cvi < 0 for all i 2 V implies the inequality cvk < 0 for k 2 S , a contradiction. 2 Proposition 6 The skeleton G(Zn) of any zonotope Zn is isometrically embeddable into the skeleton Hn of n of whose projection Zn is; n is the diameter of G(Zn).

Proof. We prove that the natural map v(S ) ! e(S ) for S 2 Sn determines an l 1

embedding of the skeleton of Zn into Hn. For this end, it is sucient to prove that the distance between vertices v(S ) and v(T ) is equal to the symmetric dierence jS
T j. We proceed by induction on m = jS
T j. The assertion is trivially true for m = 0. By the construction of Zn, each its edge which connects the vertex v(S ) with another vertex is parallel to the vector vi for some i 2 N. Suppose there is an edge which connects v(S ) with v(S 0) and is parallel to vi for i 2 S
T . Then either S 0 = S [ fig or S 0 = S ? fig, depending on i 2 T or i 2 S , respectively. Clearly, in both the cases jS 0
T j = m ? 1, and we can apply the induction step. Hence, for to prove the proposition, it is sucient to prove that v(S ) is incident to an edge parallel to vi for i 2 S
T . But this is implied by Lemma 5 if we take v(S ) as a new origin. 2 Komei Fukuda kindly permitted to us to see and to refer preliminary version of [Fuk95] containing (within his treatment of oriented matroid) a nice example of a centrally symmetric but non zonohedral polyhedron F with G(F ) ! H . The Fukuda's polyhedron contains 54 vertices, 16 hexagonal and 20 square faces. Fukuda constructed it in the dual form as a nonlinear (non-Pappus) extension of an oriented rank 3 matroid on 8 points. As a linear extension of the same matroid he obtain a dual zonohedron Z . (Z has 52 vertices, 18 hexagonal and 14 square faces.) Comparing Fukuda's picture for (duals of) Z and F , we realize that F comes from Z by the following general construction. 9

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Let C =C ;:::; and C 0=C 0 ;:::; 0 be two opposite hexagonal faces of a centrally symmetric polyhedron P , and (1; 10); :::; (6; 60) are 6 pairs of antipodal vertices of P . Denote by Q(P ) the polyhedron obtained from P by adding two new vertices v, v0 and edges (v; 1), (v; 3), (v; 5), (v0; 10), (v0; 30), (v0; 50). Clearly, G(P ) is centrally symmetric (v and v0 are antipodal) and G(P )  G(Q(P )), i.e. the skeleton of P is an isometric subgraph of the skeleton of Q(P ). If P ! Hn (it is so if P is a zonotope), then Q(P ) ! Hn also. (In fact, let a(1) = ;, a(3) = fij g, a(5) = fikg. Then a(v) = fig is uniquely determined. No other edge (v; u) is possible, since otherwise a(u) = fitg for t 6= j; k, and we get a contradiction with C being a hexagonal face.) De ne by induction Qm(P ) = Q(Qm? (P )). Let ElDo denote the elongated dodecahedron. Using Remark 2 below one can check that (combinatorially) Q(Prism ) = rhombic dodecahedron, Q (Prism ) = , Q (ElDo) = rhombic icosahedron, Q (truncated )= triacontahedron. It will be interesting to see whether Fukuda's polyhedron is minimal for parameters of non-zonohedral Q(P ) obtained from a zonohedron P . Another similar operation is as follows. Select two opposite 2m-faces C = C ;:::; m and C 0 = C 0 ;:::; m 0 . Take a path P ;i1;:::;im?2;m of length m ? 1 connecting the vertices 1 and m + 2 of C . Add new edges (ik ; k + 2), 1  k  m ? 2. Make the same operation with the face C 0. Let P denote the rhombic dodecahedron with a deleted vertex of degree 3, i.e. P is Prism with a new vertex connected to 3 nonadjacent vertices of a 6-cycle. Clearly, P is nonzonohedral polyhedron and G(P ) ! Hn (for n = 4); probably, P is minimal for those property. Also G(P ) is (one of the four) smallest nonhamiltonian polyhedral graphs realizable as Delaunay tesselations, [Dil96], Fig. 10 and 11. Other 3 such graphs of Fig. 5{12 of [Dil96] are not 5-gonal, except, 5c ! H , 8a ! H , 8b ! H , 10b ! H . Remark 1. Examples of embeddings of zonotopes into hypercubes are 1) 5 of Archimedean and their dual (Catalan) polyhedra are zonohedra: truncated ! H , truncated cuboctahedron ! H , truncated icosidodecahedron ! H , dual cuboctahedron (= rhombic dodecahedron) ! H , dual icosidodecahedron (= triacontahedron) ! H . Between their duals only the dual of the rst one has l -skeleton. 2) All 5 (combinatorially) Voronoi polyhedra are zonohedra: 3-cube=H , rhombic dodecahedron ! H , Prism ! H , elongated dodecahedron ! H , truncated ! H . 3) All 5 \golden isozonohedra" of Coxeter (zonohedra with all faces being rhombic with the diagonals in golden proportion) are: 2 types of hexahedrons (equivalent to ) ! H , rhombic dodecahedron ! H , rhombic icosahedron ! H , triacontahedron ! H . 4) Some in nite families of zonotopes are: Prism m ! Hm , m = Hm , polar zonohedra PZm ! Hm ( , rhombic dodecahedron, rhombic icosahedron for m = 3; 4; 5, respectively, see [Cox73]); PZm is not 5-gonal for m  4 (see Fig.1e for m = 5). An m-dimensional permutahedron (the Voronoi polytope of the lattice Am) ! H(m2+1). It is C , truncated for m = 2; 3, respectively. 1

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Remark 2. For an isometric m-vertex subgraph G of an n-cube (which is not an isometric subgraph of Hn? ) denote by G~ the subgraph of Hn induced by 2n ? m remaining vertices. If G is the skeleton of a zonotope Z ! Hn, then G~ can be : a)  = G if m = 2n? (examples are n? and triacontahedron); 1

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b) an isometric subgraph of Hn (for example, G~ = C for rhombic icosahedron and ~ G = C ;:::; + P ; ; ; ; + P ; ; ; ; for elongated dodecahedron (see Fig 11a); c) or not isometric subgraph of a hypercube (for example, G~ = 2K for rhombic dodecahedron, G~ = 2K for Prism and G~ is a 40-vertex non isometric subgraph of H for truncated ). Remark 3. See [DeSt96] for similar isometric embeddings of skeletons of in nite zonohedra (plane tilings) into cubic lattices Zn. For example, regular hexagonal tiling and dual Archimedean tiling [3,6,3,6] are embeddable into Z , Archimedean tiling (4,6,12) is embeddable into Z , Penrose aperiodic rhombic tiling is embeddable into Z . 10

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7 Delaunay polytopes Here we consider Delaunay polytopes of dimension at most 4 and some operations on Delaunay polytopes (see a de nition of a Delaunay polytope in x2.1). For n = 2, Delaunay polytopes have the skeletons C ! H and C = H . For n = 3, skeletons of all 5 Delaunay polytopes are also l -graphs:

= H ,  = H , ! H , Pyr ! H , Prism ! H . All 19 types of Delaunay 4-polytopes are given in [ErRy87]. The # i, 1  i  16, and the letters A,B,C below are the notations from [ErRy87]. (Elsewhere than this section, # means the number of a polyhedron in the list of 92 regular-faced polyhedra.) We get the following proposition by direct check. 1 2

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Proposition 7 1) #16= =   = H , 2) P ! H for the following P : 4

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#2=Pyr(Pyr4) with G(#2) = K2;2;1;1 , B=Pyr( 3) with G(B ) = K2;2;2;1 , C=BPyr( 3) = 4 = 21 H4; 3) P ! 12 H5 for the following P : #1=Pyr(3) = 4 , #3=Pyr(Prism3), #5, #7=1  3 = Prism(3 ), #9, #13 with G(#13) = T (5), 4) P ! 21 H6 for the following P : #10= 2  2 , #11=1  Pyr4 = Prism(Pyr4), #15= 1  3 = Prism( 3 ), A with G(A) = K6 (the cyclic 4-polytope); 5) P ! 21 H7 for P = #14 = 1  Prism3 = Prism(Prism3 ); 6) the following P are not 5-gonal: #4, #6= BPyr(Prism3), #8= Pyr( 3), #12=BPyr( 3).

Proof. Recall that a graph G is not 5-gonal if K ? K  G or K ? P ? P  G with P \ P = ;. Hence 5) above is implied by that the skeletons of #8=Pyr( ) and #12=BPyr( ) each contains the isometric subgraph K ? K . The skeletons of #4 and #6=BPyr(Prism ) contain the isometric subgraph K ? P ? P . All embeddings in 1)-4) 5

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above, except # #3,5,9, either are trivial or come from the direct product construction 23

below. The skeleton G(#k) of the four polytopes #k in 3) for k = 3; 5; 9; 13 are isometric subgraphs of the triangular graph T (5). In fact G(#3) = T (5) ? K , G(#5) = T (5) ? (v; v0), G(#9) = T (5) ? v, and G(#13) = T (5). Here T (5) ? K is the graph T (5), where any 3 mutually adjacent vertices are deleted, T (5) ? (v; v0) is T (5), where any two nonadjacent vertices are deleted, and T (5) ? v is T (5) where a vertex is deleted (vertices are deleted with incident edges). In other words, the vertices of the polytopes #k, k = 3; 5; 9; 13, are labeled by pairs ij , 1  i < j  5. For example, #3=Pyr(Prism ) has the labels #45 and i4; i5, i = 1; 2; 3. >From this embedding of Pyr(Prism ) we obtain an embedding of Pyr (Prism ) if we label the new apex by the set ;. Remark, that #13 is combinatorially equivalent to a semiregular 4-polytope ambo- (0 in the Coxeter's notations). 2 Also #8=Pyr( ) is the graph of the unit cell of the body-centered orthorhombic crystal system; other 3 kinds of this system (simple, face-centered, end face-centered) have l -graphs: , omnicapped (truncated ), 2(opposite faces)-capped . It is well known that the direct product P  P 0 of Delaunay polytopes P and P 0 is a Delaunay polytope, and G(P  P 0) = G(P )  G(P 0). Also, for every Delaunay polytope P there is a pyramid Pyr(P ) and (if P is centrally symmetric) a bipyramid BPyr(P ) which are Delaunay polytopes (see [DeGr93]). The direct product construction preserves l -ness of Delaunay polytopes. For example, G(P  P 0) ! Hm m0 if G(P ) ! Hm, G(P 0) ! Hm0 . But, the pyramid and bipyramid constructions can produce not 5-gonal Delaunay polytopes from those with l -skeletons. Consider, for example, the following Delaunay polytopes: n, n, n, n (n  5), ambo-n (n  4), the Johnson n-polytope PJ with G(PJ ) = J (n + 1; k) (3  k  b(n + 1)=2c), n?  n? for n  3; Prism ; Pyr  R . All of them have l -skeletons: see x3.1 for regular ones, also ambo-n ! Hn , PJ ! Hn , Prism ! H , Pyr ! H . The pyramid and bipyramid constructions produce from them l -polytopes in the following cases: Pyrm(n), Pyrm( n), BPyrm( n), Pyrm(Pyr ) (see x2). Additionally we have the following l -embeddings: Pyr (Prism ) ! H , Pyr(ambo-n) ! Hn , Pyr(n?  n? ) ! H n. The inclusions (i), (ii) below show that the skeletons of the mentioned there polytopes are not 5-gonal, and therefore they are not l -graphs: (i) K ? K  Pyr( n), BPyr( n), Pyr (ambo-n; n  6), Pyr(PJ ) (since K ;k  J (n + 1; k)), Pyr( Hn; n  6), Pyr (n?  n? ), BPyr(ambo-n; n  6; even), BPyr( Hn; n  6 even); (ii) K ? P ? P  BPyr (Prism ). Finally, the following skeletons are (extreme) hypermetrics (see Proposition 4.4 of [DeGr93]): Pyr( ), Pyr ( ), Pyr (ambo- ), Pyr (ambo- ), Pyr (Prism ), Pyr (Prism ). 3

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3

5

1

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1

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+1

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8 Small 1-polyhedra and examples of polytopal hypermetrics l

Combinatorial types of d-polytopes were enumerated for small values of v ? d, where v is the number of vertices; see, for example, x3.16 and Tables 1,2 on p.424 of [Gru67]. The two propositions below give l -status for polyhedra of the rst two simple classes. 1

Proposition 8 (i) For all 10 combinatorial types of polyhedra with  6 vertices, we have

a) 4 skeletons: G(3 ) = K4, G(Pyr4) = rC4, G(BPyr3 = 1-capped 3) = K5 ? e, G( 3 ) = K32 ! 1 H4; 2 b) 4 skeletons: G(Prism3) = K6 ? C6, G(Pyr5) = rC5 , K6 ? P5 , G(2-capped 3) = K6 ? P4 ! 21 H5 ; c) 2 skeletons, K6 ? P6 and K32 ? e, are not 5-gonal (each contains K5 ? P2 ? P3 ). (ii) For all 10 combinatorial types of polyhedra with  6 faces (duals of above), we have: a) 3 , Pyr4, Pyr5 and one with the skeleton K6 ? P6 are self-dual, Prism3 = BPyr3 and 3 = 3 ; b) P  ! 21 H6 if G(P ) = K6 ? P4 ; c) P  is not 5-gonal if G(P ) = K6 ? P5 , K32 ? e (Fig.6).

One can check that i-capped  , dual (i ? 1)-capped  ! Hi for 1  i  4, but dual 4-capped  (truncated  ) is not 5-gonal. Also i-capped ! Hi , 0  i  8, and dual i-capped ! Hi for 0  i  2. Compare with i-capped ! H , 0  i  6. Remark that extreme hypermetrics graphs G and G are skeletons of 4-pyramids with bases Pyr and 2-capped  , see (i) b) above. 3

3

3

1 2

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3

+6

+3 1 2

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1

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+4 1 2

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3

Proposition 9 (i) For all 5 combinatorial types of simplicial polyhedra with 7 vertices,

we have: a) 1-capped 3 ! 12 H5 , 3-capped 3 ! 12 H6 ; b) skeletons G(BPyr5), r2 P5 are not 5-gonal (they contain K5 ? P2 ? P3 , K5 ? K3 , respectively); c) skeleton rB8 of Pyr(B8) is the extreme hypermetric G4. (ii) For all 5 combinatorial types of simple polyhedra with 10 vertices (dual of above), we have: a) P  ! 21 H7 if P is a 1-capped 3 , 3-capped 3 or BPyr5 (Fig.7); b) P  is not 5-gonal if G(P ) = r2 P5 , rB8 (Fig.6).

There are 34 combinatorial types of polyhedra with 7 vertices: 2,8,11,8,5 having 6,7,8,9,10 faces, respectively. The last 5 are simplicial ones. The rst two are both not 5-gonal (Proposition 8 (ii) c)). One can enumerate isometric subgraphs (and polyhedral ones between them) of given Hn which are not isometric subgraphs of a facet of it. For example, the following graphs are such isometric polyhedral subgraphs of the Clebsh graph H : 1) C , K = G( ) (together with 2) below they are only such subgraphs of H on m  6 vertices); 1 2

1 2

5

5

4

25

5

1 2

5

2)K ? C = G(Prism ), K ? P (polyhedral), K ? P = G(2-capped  ), rC = G(Pyr ) (between all 10 6-vertex isometric subgraphs of H ); 3)  7-vertex polyhedral; #27 (1-capped Prism ), #60 (augmented Prism ), #70 (biaugmented Prism ), 1-capped , APrism ; 4) 4-polytopal: Pyr(Prism ),    , T (5), T (5) ? K , T (5) ? K , 1-capped (on a facet ) T (5); 5) 5-polytopal: Pyr (Prism ), rT (5) = G(Pyr(ambo- )). On the other hand, all hypermetric but not l -graphs with 7 vertices are known (see x7 of [DGL95]); they are 12 graphic metrics between 26 extreme hypermetrics. 6

6

3

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5

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4

1 2

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1

Proposition 10 Between all the 12 hypermetric graphs on 7 vertices, the polytopal ones are only 3-polytopal G = rB and 4-polytopal G = r C = K ? C , G = rH = K ?P . Proof. In fact, G = rH , G = rB , G , G , G have minimal degree 2. G = rB , G = rH and H = K ? P have minimal degree 3 and are not planar; so G = rH 4

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1

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16

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26

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1

is not polytopal also. Finally, G is planar and has minimal degree 3, but it can be disconnected by deleting 2 vertices. It is the skeleton of a skew (with a non-planar face) 3-polytope. 2 Metric of a graph which is hypermetric but not l -graph is necessarily an extreme hypermetric. The number of vertices of any extreme hypermetric graph is within [7,56]. Any polytope such that its skeleton is extreme hypermetric, has dimension within [3,7]. Call an extreme hypermetric graph of type I (of type II) if it generates the root lattice E (E , respectively). A graph G of type I has diameter 2, since it is an induced subgraph of the Schla i graph G(2 ) of diameter 2, and rG is an extreme hypermetric of type II. Clearly, G(Pyr(P )) = rG(P ), dim(Pyr(P ))=dimP +1. (Remark that Pyrk(C ) ! H for k 2 f0; 1g, it is extreme hypermetric for k 2 f2; 3g, and it is 7- but not 9-gonal for k = 4.) Examples of polytopal graphs G which are extreme hypermetric but are not represented as rG0 for an extreme hypermetric graph G0 are (see x5): of type I: the polyhedra ## 30, 71, 106 =M and one with skeleton G (see Fig.2); the 4-polytopes with skeletons G , G ; the 6-polytopes with skeletons K ? C = G(Pyr (Prism )), r T (5)= G(Pyr (ambo-  )), r H , the Shla i polytope 2 having 9,9,7; 7,7; 9,12,17,27 vertices, respectively. of type II: the polyhedra #37, #107= M + Pyr and the 7-polytope 3 with 10, 11 and 56 vertices, respectively. Some examples of extreme hypermetrics which are not polytopal, but are close to polytopal in some sense: a) 7-vertex graph G of type I is a planar graph of a skew polyhedron. b) The skeleton of the stella octangula (the section of by with vertices in the centers of faces of ) which is a non-convex polyhedron; it is 14-vertex graph. It contains the induced extreme hypermetric Gx. It is an isometric subgraph of the Goseet graph, and since it has diameter 3 it is of type II. c) Antiwebs AW , AW , are of type I, and AW is of type II (see x4.4(iv)). 18

1

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9 -embeddings in

1 A t-embedding of a distance kdij k is an embedding of the distance kd0ij k = kmin(t; dij k in an ln -space. In what follows, describing a t-embedding of a polyhedron P , we associate to every its vertex v a subset a(v) 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 m-path from a vertex v if there is an m-path of length m from the vertex v to a vertex of the face F . Truncated icosahedron (of diameter 9) has unique ([DeSp96]) 7-embedding into H : associate each vertex to 2+2+3 hexagons (from all 20) reachable by 0-,1-,2-paths, respectively. It is also the unique 3-embedding, but not unique 2-embedding: for example, associate every vertex to 2 its hexagons. Icosidodecahedron (of diameter 5) has unique ([DeSp96]) 4-embedding in H : associate every vertex to 1+1+2 pentagons (from all 12) reachable by 0-,1-,2-paths. It is also a unique 3-embedding, but there is another 2-embedding: associate every vertex to 2 its triangular faces. Cuboctahedron (of diameter 3) has at least two following 2-embeddings: in H : associate every vertex to its two square faces, in H : associate every vertex to its two triangular faces; this one, as well as a), b) are t-embeddings into H D P (cf. Remark 3 in x3.2). Any simple polyhedron has a 2-embedding in the tetrahedral graph J (n; 3) (so in Hn): associate every vertex to its 3 faces. If the diameter of a simple polyhedron is at least 3, then it is 3-embeddable i sizes of its faces are from f3; 4; 5g. Examples of this procedure are: 1) dodecahedron (of diameter 5) has a 3-embedding into J (12; 3); also it has a 5embedding into H , 2)  ! H (not unique), 3) it gives a 2-embedding of Prismn which turns out to be Prismn ! Hn (! H n+2 2 for even n). Another procedure: x a 5-wheel rC in the skeleton of an icosahedron and associate every vertex v to the set of all vertices of the 5-wheel at distances 0 and 1 from v; we get a (unique) 3-embedding of the icosahedron. t

l

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(

)+2

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Another interesting relaxation of our embeddings will be to consider scale-isometric embedding into hypercubes of 1-skeletons of simplicial and cubical complexes more general than the boundary complexes of polytopes. For example, the simplicial complex on f1; 2; 3; 4; 5g with the facets f1; 2; 3g, f1; 2; 4g, f1; 2; 5g has the skeleton K ? K ; the cubical complex on f1; 2; 3; 4; 5; 6g with the facets f1; 2; 3; 4g, f2; 3; 5; 6g, f1; 4; 5; 6g has the skeleton K ; . So, both are not 5-gonal. 5

33

27

3

References [AsDe80] P.Assouad and M.Deza, Espaces metriques plongeable dans hypercube: aspects combinatoires, Annals of Discrete Math. 8 (1980) 197{210. [Ass81] P.Assouad, Embeddability of regular polytopes and honeycombes in hypercubes, in: The Geometric Vein, the Coxeter Festschrift, Springer-Verlag, 1981, pp.141{147. [Bal95] A.T.Balaban, X.Liu, D.J.Klein, D.Babic, T.G.Schmalz, W.A.Seitz and M.Randic, Graph invariants for Fullerenes J.Chem. Inf. Comp. Sci. 35 (1995) 396{404. [BlBl91] G.Blind and R.Blind, The semiregular polytopes, Comment. Math. Helvetica 66 (1991) 150{154. [Ber71] M.Berman, Regular faced convex polyhedra, J. of the Franklin Institute, 291-5 (1971) 329{352. [CoSl87] J.H.Conway and N.J.A.Sloane, Sphere packings, lattices and groups, Grundlehren der mathematischen Wissenschaften, vol. 290, Springer Verlag, Berlin, 1987. [Cox73] H.S.M.Coxeter, Regular polytopes, 3d ed. Dover Publications Inc., 1973. [DDMP93] O.Delgado, A.Dress, A.Muller and M.T.Pope, Polyoxometalates:A class of compounds with remarkable topology, Molecular Engineering 3 (1993) 9{28. [Dez60] M.Deza (Tylkin), On Hamming geometry of unit cubes (in Russian), Doklady Akad. Nauk SSSR 134-5 (1960) 1037{1040. [DeGr93] M.Deza and V.P.Grishukhin, Hypermetric graphs, Quart.J.Math. Oxford (2) 44 (1993) 399{433. [DGL95] M.Deza, V.P.Grishukhin and M.Laurent, Hypermetrics in geometry of numbers, DIMACS Series in Discrete Math. and Theoretical Computer Science, vol. 20, 1995, pp.1{109. [DeSp96] M.Deza and S.Shpectorov, Recognition of l -graphs with complexity O(nm), or Football in a Hypercube, Europ. J. Combinatorics 17-2,3 (1996) [DeSt96] M.Deza and M.Stogrin, Scale-isometric embeddings of Archimedean tilings, polytopes and their duals into cubic lattices and hypercubes, in preparation. [DeTu96] M.Deza and J.Tuma, A note on l -rigid planar graphs, Europ. J. Combinatorics 17-2,3 (1996) [Dil96] M.B.Dillencourt, Polyhedra of small order and their Hamiltonian properties, J. Combinatorial Theory Ser. B 66 (1996) 87{122. [ErRy87] R.M.Erdahl and S.S.Ryshkov, The empty sphere, Canadian J. Math. 34-4 (1987) 794{824. 1

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[Fuk95] K.Fukuda, Lecture notes: A constructive approach to polyhedral geometry and mathematical programming, Institute for Mathematical Research, ETH, Zurich, Switzerland, 1995. [Gru67] B.Grunbaum, Convex Polytopes, Pure and Appl. Math., vol. 16, Interscience Publ. London{New York{Sydney, 1967. [Kel75] J.B.Kelly, Hypermetric spaces, in: The Geometry of Metric and Linear spaces, Lecture Notes in Math. 490, Springer-Verlag, 1975, pp.17{31. [KoSu92] M.Kobayasi and T.Suzuki, Calculation of coordinates of vertices of all convex polyhedra with regular faces, Bull. Univ. Electro-comm. 5(2) (1992) 147{184 (in Japanese). [Koe85] G.Koester, Note to a problem of T.Gallai and G.A.Dirac, Combinatorica 5 (1985) 227{228. [Koo90] J.Koolen, On metric properties of regular graphs, Master's Thesis, Eindhoven University of Technology, 1990. [LLR95] N.Linial, E.London and Yu.Rabinovich, The geometry of graphs and some of its algorithm applications, Combinatorica 15-2 (1995) 577{591. [Mak88] P.V.Makarov, How to deduce the 4-dimensional semiregular polytopes, Voprosy Discretnoi Geometrii, Matematich. Issledovania Inst. Matem. Akademii Nauk Moldavskoi SSR, 103 (1988) 139{150 (in Russian). [PSC90] K.F.Priskar, P.S.Soltan and V.D.Chepoi, On embeddings of planar graphs into hypercubes, Proceedings of Moldavian Academy of Sciences, Mathematics 1 (1990) 43{50 (in Russian). [Shp93] S.Shpectorov, On scale-isometric embeddings of graphs into hypercubes, Europ.J.Combinatorics 14 (1993) 117{130. [SHDC95] N.J.A.Sloane, R.H.Hardin, T.D.Du and J.H.Conway, Minimal-energy clusters of hard spheres, J. of Comp. and Discrete Geometry 14 (1995) 237{259. [Wel84] A.W. Wells, Structural Inorganic Chemistry, Oxford, 1984. [Zal69] V.A.Zalgaller, Convex polyhedra with regular faces, Seminar in Mathematics of V.A.Steklov Math. Institute, Leningrad, vol. 2; Consultants Bureau, New York, 1969.

APPENDIX

Here we give gures of skeletons of polyhedra (except the 4-polytope on Fig1f). For non hypermetric graphs we indicate on the skeleton points violating a (2k +1)-gonal inequality. Actually, it is always 5-gonal, except Fig.1c,1f). When gures show l -polyhedra, the embeddings are into Hn (except the embedding in H on Fig.11a). They are shown as follows: we label a vertex by the sequence i ; :::; im (or i ; :::; im ) if this vertex is addressed to the set fi ; :::; img (or fi ; :::; im g, respectively. Since m  15 on our gures, we use 1,...,9,0,a,...,e as addresses. 1

1 2

5

1

1

1

29

1

Fig 1. Non hypermetric polytopes: a) M = #112, triangular hebesphenorotunda, b) M = #111, bilunabirotunda, c) M = #105, snub APrism , d) #71; e) PZ ; dual rhombic icosahedron, f) Koester's graph, g) 4-polytope Pyr(icosahedron) 20

8

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4

5

Fig 2. Hypermetric non l - polyhedra: a) #30 = 1?APrism , b) #71, triaugmented Prism , c) M = #106; sphenocorona, d) rB , unique such polyhedron with  7 vertices. 1

3

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22

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30

369

46

1278 78 24

5678

68

1234

56 3456

2578 25

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3478 3468

569 5

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1235 1356

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689 8

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126 78

Fig 3. a) Dual 1?APrism ! H ; b) dual 2?APrism ! H ; c) M  ; dual tridiminished icosahedron, d) M  ; dual hebesphenomegacorona, e) M  ; dual sphenocorona, f) M  ; dual snub APrism , g)M  ; dual snub dispensoid! H ; h) M  ;dual sphenomegacorona! H 4

1 2

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Fig 4. Prismatic non-l graphs with minimal n: a) 1?M , b) 1?APrism , c) dual 1?APrism , d) 2?APrism , e) dual 2?APrism = F , f) Tow = + ; g) dual Tow 1

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1234

Fig 5. Square ortho- and gyro-bicupolas: #48 = Cup + Cup ! H and #49 = Cup + Cup ; not 5-gonal. 4

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Fig 6. Dual small not 5-gonal polyhedra P with G(P ) = K  ?e; K ?P ; r P ; rB 3

234

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467

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Fig 7. Dual small l -polyhedra: dual 1-, 2-capped , dual 2-, 3-capped  ! H ; H ; H ; H 1

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Fig 8. All l -skeletons of basic regular-faced polyhedra Mi ; 1  i  28: 1

34

Fig 9. All not hypermetric skeletons of basic regular-faced polyhedra Mi ; l  i  28:

35

1289

126789

12890a

6789 12 1234

67 35

67de

3567de

1345bc 13450abc

1345de

579a

460a 259b 12579b 469a 23580b 25ab

59

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ab

157

159a 23680b 360b 360a 159b

5

9a

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126 1

6a

34670a

580b

1345bcde

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680b

60

12340abc

1345

35de

12579a

12340a

457

6 456

236 346 15

679a 26

12

34679a

Fig 10. Fullerens: a) 1?Tow , b) dual 2?Tow , c) strained F (D h), d) F (Td), e) dual F , f) 20-bowl ! H ; g) F ! H , h) dual F (Td)! H 26

1 2

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Fig 11. a) ElDo (elongated dodecahedron)!H , b) the graph on H nElDo, c) AW ?(6; 8); antiweb AW (a hypermetric, non-l graph) 2 9

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8