Zigzag Structure of Simple Two-faced Polyhedra

Details Zigzag Structure of Simple Two-faced Polyhedra Zigzag Structure of Simple Two-faced Polyhedra

2002-12-27

arxiv.org/abs/math/0212352v2

Mathematics

Geometric Topology

33 pages, 26 figures

Scientific paper

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.

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