We consider the Goldberg-Coxeter construction $GC_{k,l}(G_0)$ (a generalization of a simplicial subdivision of the dodecahedron considered by Goldberg [Tohoku Mathematical Journal, 43 (1937) 104–108] and Coxeter [A Spectrum of Mathematics, OUP, (1971) 98–107]), which produces a plane graph from any $3$- or $4$-valent plane graph for integer parameters $k,l$. 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 central circuit in a $4$-valent plane graph $G$ is a circuit of edges, such that no two consecutive edges belong to the same face. We study the zigzag (or central circuit) structure of the resulting graph using the algebraic formalism of the moving group, the $(k,l)$-product and a finite index subgroup of $SL_2(\Bbb{Z})$, whose elements preserve the above structure. We also study the intersection pattern of zigzags (or central circuits) of $GC_{k,l}(G_0)$ and consider its projections, obtained by removing all but one zigzags (or central circuits).

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 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.

Given a lattice L, a full dimensional polytope P is called a Delaunay polytope if the set of its vertices is S ∩ L with S being an empty sphere of the lattice. Extending our previous work [DD01] on the hypermetric cone HY P 7 , we classify the six-dimensional Delaunay polytopes according to their combinatorial type. The list of 6241 combinatorial types is obtained by a study of the set of faces of the polyhedral cone HY P 7 .

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We consider polyhedral cones, associated with quasi-semi-metrics (oriented distances), in particular, with oriented multi-cuts, on n points. We computed the number of facets and of extreme rays, their adjacencies, and incidences of the cones QM ET n and OM CU T n for n = 3, 4, 5 (see Table 1) and, partially, for n = 6. Some results for general n are also given.