×

Goldberg-Coxeter construction for 3- and 4-valent plane graphs. (English) Zbl 1053.52013

Electron. J. Comb. 11, No. 1, Research paper R20, 49 p. (2004); printed version J. Comb. 11, No. 1 (2004).
Summary: We consider the Goldberg-Coxeter construction \(GC_{k,l} (G_0)\) (a generalization of a simplicial subdivision of a certain dodecahedron considered by Goldberg (1937) and Coxeter (1971), 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 \(SL_2(\mathbb{Z})\), 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).

MSC:

52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
52B10 Three-dimensional polytopes
52B15 Symmetry properties of polytopes
05C10 Planar graphs; geometric and topological aspects of graph theory

Software:

GAP
PDFBibTeX XMLCite
Full Text: EuDML EMIS