Goldberg-Coxeter Construction for 3- and 4-valent Plane Graphs

We consider the Goldberg-Coxeter construction GCk,l(G0) (a generalization of a simplicial subdivision of the dodecahedron considered in [Gold37] and [Cox71]), which produces a plane graph from any 3or 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 SL2(Z), whose elements preserve the above structure. We also study the intersection pattern of zigzags (or central circuits) of GCk,l(G0) and consider its projections, obtained by removing all but one zigzags (or central circuits).