({2,3}, 6)-spheres and their generalizations

TL;DR

This paper studies 6-regular plane graphs with faces of size 1, 2, or 3, providing enumeration methods, symmetry classifications, a new construction technique, and analyzing zigzags and circuits with bounds and classifications.

Contribution

It introduces a practical enumeration method, a new Goldberg-Coxeter construction, and classifications of tight graphs based on zigzags and circuits.

Findings

Enumerated graphs up to 53 vertices.

Classified symmetry groups of these graphs.

Established bounds and classifications for zigzags and circuits.

Abstract

We consider here 6-regular plane graphs whose faces have size 1, 2 or 3. In Section 2 a practical enumeration method is given that allowed us to enumerate them up to 53 vertices. Subsequently, in Section 3 we enumerate all possible symmetry groups of the spheres that showed up. In Section 4 we introduce a new Goldberg-Coxeter construction that takes a 6-regular plane graph G0, two integers k and l and returns two 6-regular plane graphs. Then in the final section, we consider the notions of zigzags and central circuits for the considered graphs. We introduced the notions of tightness and weak tightness for them and we prove an upper bound on the number of zigzags and central circuits of such tight graphs. We also classify the tight and weakly tight graphs with simple zigzags or central circuits.