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

### Abstract

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).