Polytopality and Cartesian products of graphs

Abstract. We study the question of polytopality of graphs: when is a given graph the graph of a polytope? We first review the known necessary conditions for a graph to be polytopal, and we provide several families of graphs which satisfy all these conditions, but which nonetheless are not graphs of polytopes. Our main contribution concerns the polytopality of Cartesian products of non-polytopal graphs. On the one hand, we show that products of simple polytopes are the only simple polytopes whose graph is a product. On the other hand, we provide a general method to construct (non-simple) polytopal products whose factors are not polytopal. Even though graphs are perhaps the most prominent feature of polytopes, we are still far from being able to answer several basic questions regarding them. For applications, one of the most important ones is to bound the diameter of the graph in terms of the number of variables and inequalities defining the polytope [San10]. From a theoretical point of view, it is striking that we cannot even efficiently decide whether a given graph occurs as the graph of a polytope or not