Bar-Magnet Polyhedra and NS-Orientations of Maps

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 non-alternation 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 existence of such orientations can be expressed algebraically by a new homology invariant of perfect matchings in the related graph of cofacial odd vertices. 1 Bar-magnet polyhedra A toy called “Roger’s Connection” provides a collection of magnetic bars and steel balls that can be used to construct polyhedra. The structures are most stable when around each vertex (a steel ball), the North and South poles of magnetic bars meeting at that vertex are alternating. This toy motivated the definition of bar-magnet polyhedra as those 3-connected plane graphs whose edges can be directed so that the directions “alternate” around each vertex, where one non-alternation is allowed if the degree of the vertex is odd. Let G be a map, i.e., a graph that is 2-cell embedded in some surface. We refer to [4] for basic definitions concerning graphs on surfaces. An NSorientation of G is an orientation of the edges so that in the clockwise order ∗Supported in part by the Ministry of Education, Science and Sport of Slovenia, Research Program P1–0507–0101.