The graph of an abstract polytope

Abstract

Recently a generalization of simple convex polytopes to combinatorial entities known as abstract polytopes has been proposed. The graph of an abstract polytope of dimension d is a regular connected graph of degree d. Given a connected regular graph Г of degree d , it is interesting to find out whether it is the graph of some abstract polytope P. We obtain necessary and sufficient conditions for this, in terms of the existence of a class of simple cycles in Г satisfying certain properties. The main result in this paper is that if a pair of simple convex polytopes or abstract polytopes have the same two-dimensional skeleton, then they are isomorphic. Every two-dimensional face of a simple convex polytope or an abstract polytope is a simple cycle in its graph. Given the graph of a simple convex polytope or an abstract polytope and the simple cycles in this graph corresponding to all its two-dimensional faces, then we show how to construct all its remaining faces. Given a regular connected graph Г and a class of simple cyles D in it, we provide necessary and sufficient conditions under which D is the class of two-dimensional faces of some abstract polytope which has Г as its graph.