Tilings of nonorientable surfaces by Steiner triple systems

Abstract

A Steiner triple system of order n (for short, STS(n)) is a system of three-element blocks (triples) of elements of an n-set such that each unordered pair of elements occurs in precisely one triple. Assign to each triple (i,j,k) ? STS(n) a topological triangle with vertices i, j, and k. Gluing together like sides of the triangles that correspond to a pair of disjoint STS(n) of a special form yields a black-and-white tiling of some closed surface. For each n ? 3 (mod 6) we prove that there exist nonisomorphic tilings of nonorientable surfaces by pairs of Steiner triple systems of order n. We also show that for half of the values n ? 1 (mod 6) there are nonisomorphic tilings of nonorientable closed surfaces.