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The graphs behind Reuleaux polyhedra

Article by Luis Montejano and Eric Pauli and Miguel Raggi and Edgardo Roldán-Pensado
  • Published in 2019
  • Added on
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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:GG (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 uτ(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 uτ(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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key
ThegraphsbehindReuleauxpolyhedra
type
article
date_added
2019-05-09
date_published
2019-05-03

BibTeX entry

@article{ThegraphsbehindReuleauxpolyhedra,
	key = {ThegraphsbehindReuleauxpolyhedra},
	type = {article},
	title = {The graphs behind Reuleaux polyhedra},
	author = {Luis Montejano and Eric Pauli and Miguel Raggi and Edgardo Rold{\'{a}}n-Pensado},
	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 {\$}\tau : G \to G^*{\$} (where {\$}G^*{\$}
is the unique dual graph), a metric mapping is a map {\$}\eta : V(G) \to \mathbb
R^3{\$} such that the diameter of {\$}\eta(G){\$} is {\$}1{\$} and for every pair of vertices
{\$}(u,v){\$} such that {\$}u\in \tau(v){\$} we have dist{\$}(\eta(u),\eta(v)) = 1{\$}. If {\$}\eta{\$}
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 \tau(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
{\$}\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 V\'azsonyi, and also to a problem of
Blaschke-Lebesgue.},
	comment = {},
	date_added = {2019-05-09},
	date_published = {2019-05-03},
	urls = {http://arxiv.org/abs/1904.12761v1,http://arxiv.org/pdf/1904.12761v1},
	collections = {Geometry},
	url = {http://arxiv.org/abs/1904.12761v1 http://arxiv.org/pdf/1904.12761v1},
	year = 2019,
	urldate = {2019-05-09},
	archivePrefix = {arXiv},
	eprint = {1904.12761},
	primaryClass = {cs.CG}
}