2002/12/27 by M. Deza, Deza, M., M. Dutour +1
Mathematics · #05C10 #52B10 #Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Primary 52B05 #Secondary 05C30 #math.CO #math.GT #msc:05C10 #msc:05C30 #msc:52B05 #msc:52B10
paper · pdf · doi:10.48550/arxiv.math/0212352
33 pages, 26 figures
arxiv created 2003/08/01 · arxiv updated 2009/11/30
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.