Bar-magnet polyhedra and NS-orientations of maps

Abstract

At the CCCG 2001 open-problem session [2], J. O'Rourke asked which polyhedra can be represented by bars and magnets. This problem can be phrased as follows: which 3-connected planar graphs may have their edges directed so that the directions "alternate" around each vertex (with one exception of nonalternation if the degree is odd). In this note we solve O'Rourke's problem and generalize it to arbitrary maps on general surfaces. Obstructions to the existence of such orientations can be expressed algebraically by a new homology invariant of perfect matchings in the related graph of cofacial odd vertices. © 2005 Springer Science+Business Media, Inc.