Fullerenes and disk-fullerenes

M. Deza   ; M. Dutour Sikirić   ; M. I. Shtogrin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 68 (2013) no. 4, pp. 665-720

Résumé

A geometric fullerene, or simply a fullerene, is the surface of a simple closed convex 3-dimensional polyhedron with only 5- and 6-gonal faces. Fullerenes are geometric models for chemical fullerenes, which form an important class of organic molecules. These molecules have been studied intensively in chemistry, physics, crystallography, and so on, and their study has led to the appearance of a vast literature on fullerenes in mathematical chemistry and combinatorial and applied geometry. In particular, several generalizations of the notion of a fullerene have been given, aiming at various applications. Here a new generalization of this notion is proposed: an $n$-disk-fullerene. It is obtained from the surface of a closed convex 3-dimensional polyhedron which has one $n$-gonal face and all other faces 5- and 6-gonal, by removing the $n$-gonal face. Only 5- and 6-disk-fullerenes correspond to geometric fullerenes. The notion of a geometric fullerene is therefore generalized from spheres to compact simply connected two-dimensional manifolds with boundary. A two-dimensional surface is said to be unshrinkable if it does not contain belts, that is, simple cycles consisting of 6-gons each of which has two neighbours adjacent at a pair of opposite edges. Shrinkability of fullerenes and $n$-disk-fullerenes is investigated. Bibliography: 87 titles.

Détail

Keywords: convex polyhedron, planar graph, fullerene, patch, disk-fullerene.
Mots-clés : polygon

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