Polytopality and Cartesian products of graphs
- 147 Downloads
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.
KeywordsRegular Graph Face Lattice Internal Vertex Circulant Graph Petersen Graph
Unable to display preview. Download preview PDF.
- [dON09]A. Guedes de Oliveira and M. Noy, Personal communication, 2009.Google Scholar
- [Pil10]V. Pilaud, Multitriangulations, pseudotriangulations and some problems of realization of polytopes, Ph.D. thesis, Université Paris 7 & Universidad de Cantabria, 2010. Available at arXiv:1009.1605.Google Scholar
- [RG96]J. Richter-Gebert, Realization spaces of polytopes, Lecture Notes in Mathematics, Vol. 1643, Springer-Verlag, Berlin, 1996.Google Scholar
- [San10]F. Santos, A counterexample to the Hirsch conjecture, Annals of Mathematics, to appear. Available at arXiv:1006.2814, 2010.Google Scholar
- [Ste22]E. Steinitz, Polyeder und Raumeinteilungen, in Encyclopädie der mathematischen Wissenschaften, Band 3 (Geometrie), Teil 3AB12, B. G. Tevbner, Leipzig, 1922, pp. 1–139.Google Scholar
- [Zie10]G. M. Ziegler, Convex polytopes: Examples and conjectures, in DocCourse Combinatorics and Geometry 2009, Part I: Intensive Courses (M. Noy and J. Pfeifle, eds.), Vol. 5.1, CRM Documents, Centre de Recerca Matemàtica, Barcelona, 2010, pp. 9–49.Google Scholar