Abstract
In a number of previous papers, techniques have been developed to construct geometric 3-configurations whose reduced Levi graph has at least one pair of parallel edges, but construction techniques for configurations whose reduced Levi graph is a simple bipartite cubic graph remained elusive. In Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration (2016), a primarily straightedge-and-compass construction, along with the construction of a certain conic section, was developed to construct configurations whose reduced Levi graph is an edge-labelled digraph with underlying graph \(K_{3,3}\). In this paper, we extend that construction to produce a construction technique for certain classes of configurations whose reduced Levi graphs are bipartite simple cubic graphs that contain at least one quadrilateral; in particular, we produce a geometric construction method that produces configurations whose reduced Levi graphs have as their underlying graphs the cyclic fence graphs, which unify even prism graphs and odd Moebius ladders.
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Research supported by a grant from the Simons Foundation (#209161 to L. Berman).
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Berman, L.W. Using conics to construct geometric 3-configurations, part II: the generalized Steiner construction. J. Geom. 108, 1055–1072 (2017). https://doi.org/10.1007/s00022-017-0394-y
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DOI: https://doi.org/10.1007/s00022-017-0394-y