Zigzag Structure of Simple Two-faced Polyhedra

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.