The graphs behind Reuleaux polyhedra
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This work is about graphs arising from Reuleaux polyhedra. Such graphs must necessarily be planar, 3-connected and strongly self-dual. We study the question of when these conditions are sufficient. If G is any such a graph with isomorphism τ : G → G^* (where G^* is the unique dual graph), a metric mapping is a map η : V(G) →R^3 such that the diameter of η(G) is 1 and for every pair of vertices (u,v) such that u∈τ(v) we have dist(η(u),η(v)) = 1. If η is injective, it is called a metric embedding. Note that a metric embedding gives rise to a Reuleaux Polyhedra. Our contributions are twofold: Firstly, we prove that any planar, 3-connected, strongly self-dual graph has a metric mapping by proving that the chromatic number of the diameter graph (whose vertices are V(G) and whose edges are pairs (u,v) such that u∈τ(v)) is at most 4, which means there exists a metric mapping to the tetrahedron. Furthermore, we use the Lovász neighborhood-complex theorem in algebraic topology to prove that the chromatic number of the diameter graph is exactly 4. Secondly, we develop algorithms that allow us to obtain every such graph with up to 14 vertices. Furthermore, we numerically construct metric embeddings for every such graph. From the theorem and this computational evidence we conjecture that every such graph is realizable as a Reuleaux polyhedron in R^3. In previous work the first and last authors described a method to construct a constant-width body from a Reuleaux polyhedron. So in essence, we also construct hundreds of new examples of constant-width bodies. This is related to a problem of Vázsonyi, and also to a problem of Blaschke-Lebesgue.
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