Abstract
We prove that the number of vertices of a polytope of a particular kind is exponentially large in the dimension of the polytope. As a corollary, we prove that an n-dimensional centrally symmetric polytope with O(n) facets has {ie1-1} vertices and that the number of r-factors in a k-regular graph is exponentially large in the number of vertices of the graph provided k≥2r+1 and every cut in the graph with at least two vertices on each side has more than k/r edges.
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References
K. Ball: Convex geometry and functional analysis, Handbook of the Geometry of Banach Spaces, Vol. I, North-Holland, Amsterdam, 2001, 161–194.
A. Barvinok: A Course in Convexity, Graduate Studies in Mathematics, 54, American Mathematical Society, Providence, RI, 2002.
A. Barvinok and A. Samorodnitsky: Random weighting, asymptotic counting, and inverse isoperimetry, Israel J. Math. 158 (2007), 159–191.
L. Esperet, F. Kardoš, A. King, D. Král and S. Norine: Exponentially many perfect matchings in cubic graphs, Adv. Math. 227 (2011), 1646–1664.
T. Figiel, J. Lindenstrauss and V.D. Milman: The dimension of almost spherical sections of convex bodies, Acta Math. 139 (1977), 53–94.
E.D. Gluskin: Extremal properties of orthogonal parallelepipeds and their applications to the geometry of Banach spaces, Mat. Sb. (N.S.) 136(178), no. 1 (1988), 85–96; translation in Math. USSR-Sb. 64 (1989), no. 1, 85–96.
L. Lovász and M. D. Plummer: Matching Theory, North-Holland Mathematics Studies, 121. Annals of Discrete Mathematics, 29; North-Holland Publishing Co., Akadémiai Kiadó, Amsterdam, Budapest, 1986.
A. Schrijver: Combinatorial Optimization. Polyhedra and Efficiency. Vol. A. Paths, Flows, Matchings, Chapters 1–38, Algorithms and Combinatorics 24, A, Springer-Verlag, Berlin, 2003
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This research was partially supported by NSF Grant DMS 0856640, by a United States-Israel BSF grant 2006377 and by ERC Advanced Grant No 267195.
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Barvinok, A. A bound for the number of vertices of a polytope with applications. Combinatorica 33, 1–10 (2013). https://doi.org/10.1007/s00493-013-2870-9
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DOI: https://doi.org/10.1007/s00493-013-2870-9