Polytopality and Cartesian products of graphs

Details Polytopality and Cartesian products of graphs Polytopality and Cartesian products of graphs

2010-09-08

arxiv.org/abs/1009.1499v1

Mathematics

Metric Geometry

21 pages, 10 figures

Scientific paper

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.

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