Face Numbers of Centrally Symmetric Polytopes Produced from Split Graphs

Ragnar Freij, Matthias Henze, Moritz W. Schmitt, Günter M. Ziegler


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.


Hansen polytopes; 3^d conjecture; Hanner polytopes; split graphs; threshold graphs

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