Polytopality and Cartesian products of graphs

TL;DR

This paper investigates the conditions under which graphs are polytopal, exploring Cartesian products of non-polytopal graphs and providing methods to construct polytopal products from non-polytopal factors.

Contribution

It offers new insights into the polytopality of Cartesian products, including characterizations of simple polytopes and a general construction method for non-simple polytopal products.

Findings

Products of simple polytopes are the only simple polytopes with polytopal graphs.

Provided a method to construct polytopal products from non-polytopal factors.

Identified families of graphs satisfying necessary conditions but not being polytopal.

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.