On a Conjecture of Micha Perles
Nagabhushana Prabhu · 2018
We prove a conjecture of Micha Perles concerning simple polytopes, for a subclass that properly contains the duals of stacked and crosspolytopes. As a consequence of a special property of this subclass it also follows that, the entire combinatorial structure of a polytope in the subclass can be recovered from its graph, by applying our results recursively. 1 Introduction Let P be a simple d-polytope and G(P ) the graph (1-skeleton) of P . Perles conjectured that every (d \\Gamma 1)-regular, induced, connected and non-separating subgraph of G(P ) determines a facet of P [2]. In this paper we prove the conjecture for a proper subclass of simple polytopes. The motivation for our results comes from two subclasses of simplicial polytopes, namely the stacked polytopes and the crosspolytopes. Polytopes obtained from a simplex by successive addition of pyramids over facets are called stacked polytopes. Stacked polytopes form an important subclass of simplicial polytopes in that, only they at...