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On Equicut Graphs
 QUARTERLY JOURNAL OF MATHEMATICS OXFORD
, 2000
"... The size sz() of an `1graph = (V; E) is the minimum of n f =tf over all the possible `1embeddings f into n f dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equic ..."
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Cited by 7 (3 self)
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The size sz() of an `1graph = (V; E) is the minimum of n f =tf over all the possible `1embeddings f into n f dimensional hypercube with scale t f . The sum of distances between all the pairs of vertices of is at most sz()dv=2ebv=2c (v = jV j). The latter is an equality if and only if is equicut graph, that is, admits an `1 embedding f that for any 1 i n f satis es x2V f(x) i 2 fdv=2e; bv=2cg. Basic properties of equicut graphs are investigated. A construction of equicut graphs from `1graphs via a natural doubling construction is given. It generalizes several wellknown constructions of polytopes and distanceregular graphs. Finally, large families of examples, mostly related to polytopes and distanceregular graphs, are presented.
Uniform Partitions of 3space, their Relatives and Embedding
 European J. of Combinatorics
, 2000
"... We review 28 uniform partitions of 3space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn. We also consider some relatives of those 28 partitions, including Achimedean 4polytopes of ConwayGuy, noncompact uniform par ..."
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Cited by 7 (2 self)
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We review 28 uniform partitions of 3space in order to find out which of them have graphs (skeletons) embeddable isometrically (or with scale 2) into some cubic lattice Zn. We also consider some relatives of those 28 partitions, including Achimedean 4polytopes of ConwayGuy, noncompact uniform partitions, Kelvin partitions and those with unique vertex figure (i.e. Delaunay star). Among last ones we indicate two continuums of aperiodic tilings by semiregular 3prisms with cubes or with regular tetrahedra and regular octahedra. On the way many new partitions are added to incomplete cases considered here. 1
Embedding of skeletons of Voronoi and Delone partitions into cubic lattices
 PROC. OF THE CONFERENCE IN HONOUR OF G.F.VORONOI
, 1997
"... The famous Deuxième mémoire of Voronoi (1908, 1909) in Crelle Journal contains, between other things, deep study of two dual partitions of R n related to an ndimensional lattice Λ. In modern terms, they are called Voronoi partition and Delone partition (Voronoi himself called the second one Lparti ..."
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Cited by 5 (4 self)
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The famous Deuxième mémoire of Voronoi (1908, 1909) in Crelle Journal contains, between other things, deep study of two dual partitions of R n related to an ndimensional lattice Λ. In modern terms, they are called Voronoi partition and Delone partition (Voronoi himself called the second one Lpartition). Both partitions coincide for the cubic lattice; we denote by Zn the skeleton of the cubic ndimensional lattice. Denote by V o(Λ), De(Λ) skeletons of Voronoi and Delone partitions for lattice Λ. So, edges of these graphs are edges of the Voronoi parallelotope and of the Delone polytopes of Λ; any minimal vector of Λ is an edge of De(Λ) but not vice versa, in general. We are interested whether infinite graph G, where G = V o(Λ) or De(Λ), either is embedded isometrically (or with doubled distances) into a Zm for some m ≥ n, or not; we use notation G → Zm or G → 1 2 Zm in the first two cases. In this note we report what we got, in this direction, for irreducible root lattices, for two generalizations of the diamond bilattice and for 3dimensional case. The validity of the following 5gonal inequality for distances is known [Dez60] to be necessary for embedding of any graph (in fact, of any metric space) into some Zm: for any vertices a, b, x, y, z we have d(a, b) + {d(x, y) + d(x, z)) + d(y, z)} ≤ ≤ {d(a, x) + d(a, y) + d(a, z)} + {d(b, x) + d(b, y) + d(b, z)}. It turns out that cases of nonembedding given in this note, came out by violation of this 5gonal inequality. Let us start with irreducible root lattices, i.e. An, Dn, En. For small dimension n, we have: De(A2) = (3 6) → 1 2 Z3, V o(A2) = (6 3) → Z3 ([As81]);
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
Embedding the graphs of regular tilings and starhoneycombs into the graphs of hypercubes and cubic lattices
, 1999
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Embedding of All Regular Tilings and StarHoneycombs
, 1998
"... We review the regular tilings of dsphere, Euclidean dspace, hyperbolic dspace and Coxeter's regular hyperbolic honeycombs (with infinite or starshaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a mcube or mdimensional cubic la ..."
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We review the regular tilings of dsphere, Euclidean dspace, hyperbolic dspace and Coxeter's regular hyperbolic honeycombs (with infinite or starshaped cells or vertex figures) with respect of possible embedding, isometric up to a scale, of their skeletons into a mcube or mdimensional cubic lattice. In section 2 the last remaining 2dimensional case is decided: for any odd m 7, starhoneycombs m m 2 are embeddable while m 2 m are not (unique case of nonembedding for dimension 2). As a spherical analogue of those honeycombs, we enumerate, in section 3, 36 Riemann surfaces representing all nine regular polyhedra on the sphere. In section 4, nonembeddability of all remaining starhoneycombs (on 3sphere and hyperbolic 4space) is proved. In the last section 5, all cases of embedding for dimension d ? 2 are identified. Besides hypersimplices and hyperoctahedra, they are exactly those with bipartite skeleton: hypercubes, cubic lattices and 8, 2, 1 tilings of hyperbolic 3, ...
Isometric Embedding of Mosaics Into Cubic Lattices
, 2001
"... We review mosaics T (tilings of Euclidean plane by regular polygons) with respect to possible embedding, isometric up to a scale, of their skeletons or the skeletons of their duals T , into some cubic lattice Z n . The main result of this paper is the classication, given in Table 1, of all 58 suc ..."
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We review mosaics T (tilings of Euclidean plane by regular polygons) with respect to possible embedding, isometric up to a scale, of their skeletons or the skeletons of their duals T , into some cubic lattice Z n . The main result of this paper is the classication, given in Table 1, of all 58 such mosaics among all 165 mosaics of the catalog, given in [Cha89] and including all main classications of mosaics. We consider mosaics T (i.e. edgetoedge planar tilings by regular polygons), such that the skeleton graph of T or of T embeds isometrically, up to a scale , into the skeleton of ndimensional cubic lattice Z n for some n 1. Such embedding will be denoted by T ! 1 Z n and T ! 1 Z n . For planar embeddable graph, we have = 1; 2 and such embedding, if the graph does not contain K 4 , is essentially unique; see [CDG97]. The following examples illustrate those notions. Example 1. A path P n (with n vertices) embeds into an hypercube H n 1 , as well as into...