Russian Math. Surveys 68:4 665ҽ720 Uspekhi Mat . Nauk 68:4 69ҽ128 DOI 10.1070/RM2013v068n04ABEH004850 To the memory of Boris Nikolaevich Delaunay Fullerenes and disk-fullerenes M. Deza, M. Dutour Sikiri´ c, and M. I. Shtogrin Abstract. 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 stud- ied intensively in chemistry, physics, crystallography, and so on, and their study has led to the appearance of a vast literature on fullerenes in math- ematical chemistry and combinatorial and applied geometry. In particu- lar, 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 geo- metric 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. Keywords: polygon, convex polyhedron, planar graph, fullerene, patch, disk-fullerene. Contents Introduction 666 1. Convex polyhedra 668 2. Fullerenes and disk-fullerenes 671 2.1. Fullerenes and abstract fullerenes 671 2.2. Disk-fullerenes and abstract disk-fullerenes 677 2.3. The structure of fullerenes and disk-fullerenes 687 2.4. Unshrinkable fullerenes and disk-fullerenes 694 The second author was supported by a grant from the Alexander von Humboldt Founda- tion. The third author was supported by grant no. 11.G34.31.0053 from the Government of the Russian Federation, by the Programme for the Support of Leading Scientiic Schools (grant no. НШ-4995.2012.1), and by the Russian Foundation for Basic Research (grant no. 11-01-00633). AMS 2010 Mathematics Subject Classification. Primary 52A15, 57M20, 05C10. c ⃝ 2013 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.