In this paper, we enumerate the parameter matrices of all perfect $2$-colorings of the Platonic graphs consisting of the tetrahedral graph, the cubical graph, the octahedral graph, the dodecahedral graph, and  the icosahedral graph.

We define a nearly platonic graph to be a finite k-regular simple planar graph in which all but a small number of the faces have the same degree. We show that it is impossible for such a graph to have exactly one disparate face, and offer some conjectures, including the conjecture that nearly platonic graphs with two disparate faces come in a small set of families.

We call a 3-manifold Platonic if it can be decomposed into isometric Platonic solids. Generalizing an earlier publication by the author and others where this was done in case of the hyperbolic ideal tetrahedron, we give a census of hyperbolic Platonic manifolds and all of their Platonic tessellations. For the octahedral case, we also identify which manifolds are complements of an augmented knot...

d-spheres are defined graph theoretically and inductively as the empty graph in dimension d = −1 and d-dimensional graphs for which all unit spheres S(x) are (d−1)-spheres and such that for d ≥ 0 the removal of one vertex renders the graph contractible. Eulerian d-spheres are geometric d-spheres which can be colored with d + 1 colors. They are Eulerian graphs in the classical sense and for d ≥ ...

In our work on robotic manipulation, we required a method for determining the orientation of a marked sphere from a single visual image. Our solution utilizes features of different colors painted on the sphere at the vertices of the Platonic solids. The main result of this paper is the minimization of the number of feature colors needed to solve the correspondence problem. The minimization of c...

We develop a combinatorial method to show that the dodecahedron graph has, up to rotation and reflection, a unique Hamiltonian cycle. Platonic graphs with this property are called topologically uniquely Hamiltonian. The same method is used to demonstrate topologically distinct Hamiltonian cycles on the icosahedron graph and to show that a regular graph embeddable on the 2-holed torus is topolog...

A k -regular planar graph G is nearly Platonic when all faces but one are of the same degree while remaining face a different degree. We show that no such graphs with connectivity can exist. This complements recent result by Keith, Froncek, and Kreher on non-existence 2-connected graphs.

We construct sphaleron solutions with discrete symmetries in Yang-Mills-Higgs theory coupled to a dilaton. These platonic sphalerons are related to rational maps of degree N . We demonstrate that, in the presence of a dilaton, for a given rational map excited platonic sphalerons exist beside the fundamental platonic sphalerons. We focus on platonic sphaleron solutions with N = 4, which possess ...

We give a decomposition theorem for Platonic graphs over finite fields and use this to determine the spectrum of these graphs. We also derive estimates for the isoperimetric numbers of the graphs. THE SPECTRUM OF PLATONIC GRAPHS OVER FINITE FIELDS MICHELLE DEDEO, DOMINIC LANPHIER, AND MARVIN MINEI Abstract. We give a decomposition theorem for Platonic graphs over finite fields and use this to d...