Cubic inflation, mirror graphs, regular maps, and partial cubes

Authors:
Bostjan Brešar;Sandi Klavžar;Alenka Lipovec;Bojan Mohar
Affiliations:
University of Maribor, FEECS, Smetanova 17, 2000 Maribor, Slovenia;Department of Mathematics, University of Maribor, Koroska cesta 160, 2000 Maribor, Slovenia;Department of Education, University of Maribor, Koroska cesta 160, 2000 Maribor, Slovenia;Department of Mathematics, University of Ljubljana, 1000 Ljubljana, Slovenia
Venue:
European Journal of Combinatorics
Year:
2004

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Abstract

Partial cubes are, by definition, isometric subgraphs of hypercubes. Cubic inflation is an operation that transforms a 2-cell embedded graph G into a cubic graph embedded in the same surface; its result can be described as the dual of the barycentric subdivision of G. New concepts of mirror and pre-mirror graphs are also introduced. They give rise to a characterization of Platonic graphs (i) as pre-mirror graphs and (ii) as planar graphs of minimum degree at least three whose cubic inflation is a mirror graph. Using cubic inflation we find five new prime cubic partial cubes.