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Clin d'Oeil on L_1Embeddable Planar Graphs
, 1996
"... In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many importa ..."
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Cited by 19 (2 self)
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In this note we present some properties of L1embeddable planar garphs. We show that every such graph G has a scale 2 embedding into a hypercube. Further, under some additional conditions we show that for a simple circuit C of G the subgraph H of G bounded by C is also L1embeddable. In many important cases, the length of C is the dimension of the smallest cube in which H has a scale embedding. Using these facts we establish the L1embeddability of a list of planar graphs.
The Skeleton of the 120cell is not 5gonal
, 1996
"... The edgegraphs of mpolygons and of 5 Platonic solids admit (unique) isometric embedding into a half cube (of dimension m and 6, resp.). The skeletons of ff n , fi n , fl n are also ` 1 graphs but it was proved in [1] that 24cell and 600cell are not (since not 5 and not 7gonal, resp.). We sho ..."
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Cited by 5 (4 self)
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The edgegraphs of mpolygons and of 5 Platonic solids admit (unique) isometric embedding into a half cube (of dimension m and 6, resp.). The skeletons of ff n , fi n , fl n are also ` 1 graphs but it was proved in [1] that 24cell and 600cell are not (since not 5 and not 7gonal, resp.). We show here that the last remaining regular polytope, 120cell, is also not ` 1 graph.
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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Cited by 2 (0 self)
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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.