Polytopality and Cartesian products of graphs

Julian Pfeifle (1) , Vincent Pilaud (2) , Francisco Santos (3)

1 UPC - Universitat Politècnica de Catalunya = Université polytechnique de Catalogne [Barcelona]
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]
3 MATESCO - Departamento de Matemáticas, Estadística y Computación

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 present three 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.

Mots clés: Polytopes; Cartesian product

Domaines: Combinatoire [math.CO]

Julian Pfeifle, Vincent Pilaud, Francisco Santos. Polytopality and Cartesian products of graphs. Israel Journal of Mathematics, 2012, 192 (1), pp.121-141. ⟨10.1007/s11856-012-0049-5⟩. ⟨hal-00778099⟩