A bound for the number of vertices of a polytope with applications

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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Correspondence to Alexander Barvinok.

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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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Mathematics Subject Classification (2000)

  • 52B12
  • 05A16
  • 05C70
  • 05C30