Polytopality of Maniplexes

2016/04/05 by Garza-Vargas, Jorge, Hubard, Isabel · 1 citation
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1604.01164

Abstract

Given an abstract polytope \cal P, its flag graph is the edge-coloured graph whose vertices are the flags of \cal P and the i-edges correspond to i-adjacent flags. Flag graphs of polytopes are maniplexes. On the other hand, given a maniplex \cal M, on can define a poset \cal P_M by means of the non empty intersection of its faces. In this paper we give necessary and sufficient conditions (in terms of graphs) on a maniplex \cal M in order for \cal P_M to be an abstract polytope. Moreover, in such case, we show that \cal M is isomorphic to the flag graph of \cal P_M. This in turn gives necessary and sufficient conditions for a maniplex to be (isomorphic to) the flag graph of a polytope.

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