ИСТИНА

This is joint work with Victor Buchstaber. A Pogorelov polytope (Pog-polytope) is a combinatorial simple convex 3-polytope that has a bounded right-angled realization in the Lobachevsky (hyperbolic) space L^3. Theorems by A.V.Pogorelov (1967) and E.M.Andreev (1970) imply that such polytopes are characterised by the condition that they are different from the 3-simplex and have no 3- and 4-belts, where a k-belt is a cyclic sequence of facets with empty common intersection such that facets are adjacent if and only if they follow each other. This condition appeared in the work by G.~D.~Birkhoff (1913), who proved that the 4 Colors Conjecture could be proved only for this class of polytopes and even for a smaller class with an additional restriction that any 5-belt surrounds a facet (Pog*-polytopes). In graph theory graphs of Pog and Pog*-polytopes are known as planar cyclically and strongly cyclically 5-edge-connected (c5- and c*5-connected). In 1974 D.~Barnette and J.W.Butler proved that any c5-connected planar graph is obtained from the graph of the dodecahedron by a sequence of operations of 3 types: an addition of an edge (A_1), a subdivision of a pentagon (A_2), and an addition of a pair of edges (A_3). A k-barrel B_k is a 3-polytope with the surface glued from two disks consisting of a k-gon surrounded by 5-gons. B_k is obtained from B_{k-1} by A_3. In 1977 D.Barnette proved that any c*5-connected planar graph is obtained from the graph of some B_k, k>=5, by a sequence of operations A_1, and any c5-connected planar graph is obtained from some B_k, k>=5, by a sequence of operations A_1 and A_2. There is a special case of A_1 when we cut off two adjacent edges of a polytope by one plane (a (2,k)-truncation, where k is the number of edges of a face spanned by the two edges). Combining methods by D.Barnette and the authors we obtain. Theorem 1. [2,3] A simple 3-polytope P is Pog if and only if either P=B_k, k>=5, or P is obtained from B_5 or B_6 by a sequence of (2,k)-truncations, k>= 6, and operations A_2. It is a Pog*-polytope if and only if either P=B_k, k>=5, or P is obtained from B_6 by a sequence of (2,k)-truncations, k>=6. Results by T.Doslic (1998, 2003) imply that any fullerene (a simple 3-polytope with only 5- and 6-gonal faces) is a Pog-polytope. Results by F.Kardos, F.Skrekovski and K.Kutnar, D.Marusic (2008) imply that a fullerene is not Pog* if and only if it is a (5,0)-nanotube (i.e. is obtained from B_5 by a sequence of A_3 applied to a 5-gon surrounded by 5-gons). Denote by F_{5,<=7} the set of all simple 3-polytopes with 5-, 6- and at most one 7-gon, where the 7-gon is adjacent to a 5-gon. These polytopes are Pog ([2]). Theorem 2.[2] Any fullerene different from a (5,0)-nanotube can be obtained from B_6 by a sequence of (2,6) and (2,7)-truncations such that intermediate polytopes belong to F_{5,<=7}. Theorem 3. [3] A polytope in F_{5,<=7} is not Pog* iff P is not equal to B_5 and P contains a 5-gon surrounded by 5-gons. Such a polytope is obtained from a fullerene by a sequence of A_2. Any other polytope P in F_{5,<= 7} is obtained from B_6 by a sequence of (2,6)- and (2,7)-truncations, and operations O_1,O_2,O_3, where O_i are certain compositions of these truncations, such that intermediate polytopes also belong to F_{5,<=7}. Acknowledgments. The work was partially supported by the RFBR grant No 17-01-00671. Bibliography: [1]V.M.Buchstaber, N.Yu.Erokhovets, Fullerenes, Polytopes and Toric Topology, in: Combinatorial and Toric Homotopy: Introductory Lectures of Lecture Notes Series, Institute for Mathematical Sciences, National University of Singapore, World Scientific Publishing Co., Singapore, 2017, Volume 35, 67--178. [2]{V.M.Buchstaber, N.Yu.Erokhovets, Constructions of families of three-dimensional polytopes, characteristic patches of fullerenes, and Pogorelov polytopes. Izvestiya: Mathematics, 81:5 (2017) 901--972. [3] N.Yu.Erokhovets, Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face. Symmetry 10(3) (2018), 67.