  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.
