Symmetric colorings of polypolyhedra

TL;DR

This paper studies symmetric edge colorings of polypolyhedra, classifies their coloring patterns, and counts the number of such colorings, revealing connections to matchings and visual bands on these polyhedral models.

Contribution

It provides a comprehensive enumeration of symmetric edge colorings of polypolyhedra, including new proofs and classifications linking colorings to matchings and visual bands.

Findings

Counted the number of symmetric colorings for each polypolyhedron.

Established correspondence between colorings and matchings on the dodecahedron graph.

Identified visual-band colorings for certain non-polygon-component polypolyhedra.

Abstract

Polypolyhedra (after R. Lang) are compounds of edge-transitive 1-skeleta. There are 54 topologically different polypolyhedra, and each has icosidodecahedral, cuboctahedral, or tetrahedral symmetry, all are realizable as modular origami models with one module per skeleton edge. Consider a coloring in which each edge of a given component receives a different color, and where the coloring (up to global color permutation) is invariant under the polypolyhedron's symmetry group. On the Five Intersecting Tetrahedra, the edges of each color form visual bands on the model, and correspond to matchings on the dodecahedron graph. We count the number of such colorings and give three proofs. For each of the non-polygon-component polypolyhedra, there is a corresponding matching coloring, and we count the number of these matching colorings. For some of the non-polygon-component polypolyhedra, there is a corresponding visual-band coloring, and we count the number of these band colorings.