Polytopal complexes realizing products of graphs

When is the Cartesian product of two graphs the graph of a polytope, of a cellular sphere, or even of a combinatorial manifold? In this note, we determine all 3-polytopal complexes whose graph is the Cartesian product of a 3-cycle by a Petersen graph. Through this specific example, we showcase certain techniques which seem relevant to enumerate all polytopal complexes realizing a given product. In this note, we investigate the question of finding polytopes (or more generally polytopal complexes) with a prescribed graph. This harks back to Steinitz’s Theorem [1], which characterizes the graphs of 3-polytopes as the 3-connected planar graphs, and thus ensures that 3-polytopality is polynomially decidable. In contrast, Richter-Gebert proved that deciding 4-polytopality is NP-hard, as a consequence of his work on realization spaces of 4-polytopes [2]. Motivated by this computational threshold, we focus on deciding 4-polytopality for the subclass of Cartesian products of graphs. Polytopality of Cartesian products of graphs was initially studied in [3]. By definition, polytopality is preserved by taking products: the graph of a product of polytopes is the product of their graphs. We are interested in the reciprocal question: can a product of non-polytopal graphs be polytopal? The answer differs significantly according to whether we require the realizing polytope to be simple or not [3, Theo. 2.2 and Prop. 2.7]: (1) A Cartesian product of regular graphs is the graph of a simple polytope if and only if its factors are. (2) There exist (non-simple) polytopal products of non-polytopal regular graphs. This note studies the 4-polytopality of the product of a cycle by a (small) nonpolytopal 3-regular graph, for which the above-mentioned results do not apply. Focussing on the product K3 × Pet of a 3-cycle by a Petersen graph, we illustrate several useful techniques to understand 4-polytopality of Cartesian products in general. Our approach consists in enumerating all 3-polytopal complexes whose graph is K3 × Pet, and requires two steps: we first compute all possible facets (3-dimensional faces) of all possible 3-polytopal complex realizing K3×Pet, and we then study all possible ways to glue these facets along ridges (2-dimensional faces) to form a complex with the desired graph. Supported by Netherlands Organization for Scientific Research (NWO) Vidi grant 639.032.917. Supported by mec grants MTM2008-03020 and MTM2009-07242, and agaur grant 2009 SGR 1040. Supported by mec grant MTM2008-04699-C03-02. CRM Documents, vol. 8, Centre de Recerca Matemàtica, Bellaterra (Barcelona), 2011 1 2 Polytopal complexes realizing products of graphs