Papers › The graphs behind Reuleaux polyhedra
The graphs behind Reuleaux polyhedra
Luis Montejano, Eric Pauli, Miguel Raggi, Edgardo Roldán-Pensado
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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) →ℝ³ 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$(\eta(u),\eta(v)) = 1$. 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 Lov\'asz 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 ℝ³. 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 V\'azsonyi, and also to a problem of Blaschke-Lebesgue.
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