Zigzag Structure of Simple Two-faced Polyhedra

TL;DR

This paper studies zigzag and railroad structures in 3-valent plane graphs, especially simple two-faced polyhedra, providing complete descriptions, classifications, bounds, and constructions for various face configurations.

Contribution

It offers a comprehensive analysis of zigzag structures in simple two-faced polyhedra, including classifications, symmetry groups, bounds, and explicit constructions for different face types.

Findings

Complete description of zigzag structures for (3,6) faces.

Classification of tight graphs with simple zigzags for (4,6) faces.

Upper bound of 9 zigzags in tight graphs.

Abstract

A zigzag in a plane graph is a circuit of edges, such that any two, but no three, consecutive edges belong to the same face. A railroad in a plane graph is a circuit of hexagonal faces, such that any hexagon is adjacent to its neighbors on opposite edges. A graph without a railroad is called tight. We consider the zigzag and railroad structures of general 3-valent plane graph and, especially, of simple two-faced polyhedra, i.e., 3-valent 3-polytopes with only $a$-gonal and $b$-gonal faces, where $3 \le a < b \le 6$; the main cases are $(a,b)=(3,6)$, $(4,6)$ and $(5,6)$ (the fullerenes). We completely describe the zigzag structure for the case $(a,b)$=$(3,6)$. For the case $(a,b)$=$(4,6)$ we describe symmetry groups, classify all tight graphs with simple zigzags and give the upper bound 9 for the number of zigzags in general tight graphs. For the remaining case $(a,b)$=$(5,6)$ we give a construction realizing a prescribed zigzag structure.