The weakly neighborly polyhedral maps on the torus (Q1059864)

A triangulation of a surface is called neighborly if each pair of vertices is contained in some triangle. The authors call a polyhedral map on a surface (i.e. a cellcomplex consisting of p-gons as facets, \(p\geq 3)\) weakly neighborly if each pair of vertices lies in a common facet. They show that there are exactly five nonisomorphic weakly neighborly polyhedral maps on a orientable surface of genus 1. An a priori calculation for the possible p-vectors shows that the number of vertices has to lie between 7 and 10. Only the cases of 7,8,9 vertices actually occur. Three of these cell decompositions can be realized by polyhedra in Euclidean 3-space. These are Császár's torus with 7 vertices, the product decomposition with \(3\cdot 3=9\) vertices and a quite irregular case with 8 vertices. For the corresponding problem in the case of surfaces of higher genus see the authors [J. Comb. Theory, Ser. A, to appear, and Isr. J. Math., to appear].