Face Numbers of Centrally Symmetric Polytopes Produced from Split Graphs

  • Ragnar Freij
  • Matthias Henze
  • Moritz W. Schmitt
  • G√ľnter M. Ziegler
Keywords: Hansen polytopes, 3^d conjecture, Hanner polytopes, split graphs, threshold graphs


We analyze a remarkable class of centrally symmetric polytopes, the Hansen polytopes of split graphs. We confirm Kalai's $3^d$ conjecture for such polytopes (they all have at least $3^d$ nonempty faces) and show that the Hanner polytopes among them (which have exactly $3^d$ nonempty faces) correspond to threshold graphs. Our study produces a new family of Hansen polytopes that have only $3^d+16$ nonempty faces.
Article Number