Zigzag Structures of Simple Two-Faced Polyhedra

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 ≤ a < b ≤ 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. Mathematics Subject Classification. Primary 52B05, 52B10; Secondary 05C30, 05C10.