Nonrealizable Minimal Vertex Triangulations of Surfaces: Showing Nonrealizability Using Oriented Matroids and Satisfiability Solvers

We show that no minimal vertex triangulation of a closed, connected, orientable 2-manifold of genus 6 admits a polyhedral embedding in R. We also provide examples of minimal vertex triangulations of closed, connected, orientable 2-manifolds of genus 5 that do not admit any polyhedral embeddings. We construct a new infinite family of non-realizable triangulations of surfaces. These results were achieved by transforming the problem of finding suitable oriented matroids into a satisfiability problem. This method can be applied to other geometric realizability problems, e.g. for face lattices of polytopes. Grünbaum conjectured [15, Exercise 13.2.3] that all triangulated surfaces (compact, orientable, connected, 2-dimensional manifolds without boundary) admit polyhedral embeddings in R. This conjecture was shown to be false by Bokowski and Guedes de Oliveira [5]. They showed that one special triangulation with 12 vertices of a surface of genus 6 does not admit a polyhedral embedding in R. Recently, Archdeacon et al. [2] settled the case of genus 1 by showing that all triangulations of the torus admit a polyhedral embedding. Still, triangulated surfaces with polyhedral embeddings can be quite complicated. McMullen, Schulz, and Wills constructed polyhedral embeddings of triangulated surfaces with n vertices of genus Θ(n logn) ([23], see also [30]). However, a gap remains: Jungerman and Ringel [18, 27] showed that n vertices suffice to triangulate a surface of genus Θ(n) and explicitly constructed such triangulations. So, can we construct polyhedral embeddings of triangulated surfaces with few vertices? In the case of 2-spheres the combinatorial bound is sharp; this is a consequence of Steinitz’s Theorem [28]. It is known that all vertex minimal triangulations of surfaces up to genus 4 admit polyhedral embeddings (genus 1 was first done by Császár [11], the 2000 Mathematics Subject Classification. Primary 52B70; Secondary 52C40. The author was supported by a scholarship of the Deutsche Telekom Foundation.