Symmetric colorings of polypolyhedra

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

Parsimonious Edge-coloring on Surfaces

We correct a small error in a 1996 paper of Albertson and Haas, and extend their lower bound for the fraction of properly colorable edges of planar subcubic graphs that are simple, connected, bridgeless, and edge-maximal to other surface embeddings of subcubic graphs

Triangle-free Uniquely 3-Edge Colorable Cubic Graphs

This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs. The only such graph previously known was given by Tutte in 1976

Triangle-free Uniquely 3-Edge Colorable Cubic Graphs

This paper presents infinitely many new examples of triangle-free uniquely 3-edge colorable cubic graphs.  The only such graph previously known was given by Tutte in 1976

Journal for Mathematics and the Arts Call for Papers: Special Issue on the Mathematics of Fiber Arts

The Journal for Mathematics and the Arts (https://www.tandfonline.com/journals/tmaa20) a peer-reviewed journal that focuses on connections between mathematics and the arts, is pleased to announce a call for papers for a special issue on Mathematics of Fiber Arts. Please send your queries via email to the guest editors. Initial submission of complete manuscripts is due August 1, 2022. The issue is currently scheduled to appear in Fall 2023

Counting Centralizers in Finite Groups

We discuss various results on the number of commuting pairs and the sizes of the centralizers of a group