The graphs behind Reuleaux polyhedra

Abstract

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 $τ: G \to G^*$ (where $G^*$ is the unique dual graph), a metric mapping is a map $η: V(G) \to \mathbb R^3$ such that the diameter of $η(G)$ is $1$ and for every pair of vertices $(u,v)$ such that $u\in τ(v)$ 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 $u\in τ(v)$) 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 $\mathbb R^3$. 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.