Far Eastern Mathematical Journal

To content of the issue

k-belts and edge-cycles of three-dimensional simple polytopes with at most hexagonal facets

Erokhovets N.Yu.

2015, issue 2, . 197-213

We describe the structure of k-belts on simple 3-polytopes with at most hexagonal facets. As a corollary we prove that the number of patches that can be bounded by a simple edge-cycle of given length on such polytopes different from nanotubes, is finite.

k-belt, simple edge-cycle, patch, cyclic edge-cut, three-dimensional polytope, fullerene

Download the article (PDF-file)


[1] T. Doslic, "Cyclical edge-connectivity of fullerene graphs and (k, 6)-cages", Journal of Mathematical Chemistry, 33:2 (2003), 103-112.
[2] A.R. Ashrafi, Z. Mehranian, "Topological Study of (3, 6) - and (4, 6)-Fullerenes", Topological Modelling of Nanostructures and Extended Systems, Series Carbon Materials: Chemistry and Physics. V. 7, 2013, 487-510.
[3] G. Brinkman, A.W. Dress, "A constructive enumaretation of fullerens", J.Algorithms, 23:2 (1997), 345-358.
[4] G. Brinkmann, J. Goedgebeur and B. D. McKay, "The Generation of Fullerenes", Journal of Chemical Information and Modeling, 52:11 (2012), 2910-2918.
[5] M. Endo, H.W. Kroto, "Formation of carbon nanofibers", J. Phys. Chem., 96 (1992), 6941-6944.
[6] G. Brinkmann, G. Caporossi and P. Hansen, "A survey and new results on computer enumeration of polyhex and fusene hydrocarbons", J. Chem. Inf. Comput. Sci., 43 (2003), 842-851.
[7] G. Brinkmann, O. Delgado Friedrichs and U. von Nathusius, "Numbers of faces and boundary encodings of patches", Graphs and discovery, DIMACS Ser. Discrete Math. Theoret. Comput. Sci., 69 (2005), 27-38.
[8] G. Brinkmann, J.E. Graver and C. Justus, "Numbers of Faces in Disordered Patches", J. Math. Chem., 45 (2009), 263-278.
[9] J. Bornhoft, G. Brinkmann, J. Greinus, "Pentagon-hexagon-patches with short boundaries", European Journal of Combinatorics, 24 (2003), 517-529.
[10] M. Hasheminezhad, H. Fleischner, B.D. McKay, "A universal set of growth operations for fullerenes", Chemical Physics Letters, 464 (2008), 118-121.
[11] V.M. Buchstaber, N.Yu. Erokhovets, "Construction of fullerenes", arXiv 1510.02948v1[math.CO], 2015.
[12] J.E. Graver, CM. Graves, "Fullerene patches I", Ars Mathematica Contemporanea, 3 (2010), 109-120.
[13] J.E. Graver, C.M. Graves, S.J. Graves, "Fullerene patches II", Ars Mathematica Contemporanea, 7 (2004), 405-421.
[14] T. Doslis, "On lower bounds of number of perfect matchings in fullerene graphs", Journal of Mathematical Chemistry, 24 (2008), 359-364.
[15] K. Kutnar, D. Marusis, "On cyclic edge-connectivity of fullerenes", Discr. Appl. Math., 156 (2008), 1661-1669.
[16] F. Kardos, R. Skrekovski, "Cyclic edge-cuts in fullerene graphs", J. Math. Chem, 22 (2008), 121-132.
[17] F. Kardos M. Krnc, B. Luzar, R. Skrekovski, "Cyclic 7-edge-cuts in fullerene graphs", Journal of Mathematical Chemistry, 47:2 (2010), 771-789.
[18] M. Deza, M. Djutur Sikirich, M.I. Shtogrin, "Fullereny i disk-fullereny", UMN, 68:4(412) (2013), 69-128.
[19] P. Schwerdtfeger, L.N. Wirz and J. Avery, "The topology of fullerenes", WIREs Comput Mol Sci, 2015 doi:10.1002/wcms.1207.
[20] Je.Je. Lord, A.L. Makkej, S. Ranganatan, Novaja geometrija dlja novyh materialov, Fizmatlit, M, 2010.
[21] B. Griinbaum, Convex polytopes (Graduate texts in Mathematics 221), Springer-Verlag, New York, 2003.
[22] G.M. Cigler, Teorija mnogogrannikov, MCNMO, M, 2014.
[23] V.M. Buhshtaber, N.Ju. Erohovec, "Usechenija prostyh mnogogrannikov i prilozhenija", Trudy MIAN im. V.A. Steklova, 289 (2015), 115-144.

To content of the issue