Zigzag Structures of Simple Two-Faced Polyhedra

Source abstract

A zigzag in a plane graph is a circuit of edges, such that any two, but not 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 neighbours 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 $3 \leq a < b \leq 6$ ; 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.