ISSN 0032-9460, Problems of Information Transmission, 2007, Vol. 43, No. 3, pp. 213–224.
Pleiades Publishing, Inc., 2007.
Original Russian Text
F.I. Solov’eva, 2007, published in Problemy Peredachi Informatsii, 2007, Vol. 43, No. 3, pp. 54–65.
Tilings of Nonorientable Surfaces
by Steiner Triple Systems
F. I. Solov’eva
Sobolev Institute of Mathematics, Siberian Branch of the RAS, Novosibirsk
Novosibirsk State University
Received March 12, 2007; in ﬁnal form, May 17, 2007
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 pre-
cisely 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(mod6)
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(mod6)thereare
nonisomorphic tilings of nonorientable closed surfaces.
The paper studies two-colored tilings of nonorientable closed surfaces by pairs of Steiner triple
systems of a special form. Two-colored tilings are also interpreted as two-colored triangular em-
beddings of a complete graph in closed surfaces (see, e.g., [1–4]). In our opinion, the term tiling
(as will be seen from deﬁnitions given below) is more appropriate for the situation in question.
A long and excited way preceded the solution of the famous Heawood problem given by Ringel and
Youngs in 1968. This problem concerns the investigation of triangular embeddings (not necessar-
ily two-colored) of a complete graph in closed surfaces. The study of triangular embeddings of a
complete graph in closed surfaces (and in closed pseudo-surfaces) gave rise to many new directions
related to the graph embedding problem in the theory of block designs, graph theory, combinato-
rial topology, algebra, and classical geometry (see survey  and also [1, 2, 4]). Despite numerous
results on embeddings of a complete graph K
into closed surfaces, there is a series of unsolved
problems. For instance, it is unknown whether there are two-colored tilings of nonorientable closed
surfaces with (n − 3)(n − 4)/6crosscapsforeachn ≡ 1(mod6)(notallcasesofthesolution
of the Heawood problem given by Ringel and Youngs in , see also , give such tilings). For
an arbitrary Steiner triple system of order n ≡ 1 or 3 (mod 6), does there exists another Steiner
system of the same order corresponding to it such that both systems give a tiling (orientable or
nonorientable) of a closed surface? Of interest is also enumeration of all nonisomorphic two-colored
tilings of orientable (nonorientable) surfaces of a given genus. Other unsolved problems can be
found in . Recall that a Steiner triple system of order n (for short, STS (n)) is a system of
blocks of length three (triples) of elements of an n-set such that each unordered pair of elements
occurs in precisely one triple. For basic deﬁnitions of combinatorial topology, in particular, deﬁ-
nitions of a closed surface, genus of a surface, orientable and nonorientable manifolds, a crosscap,
see, e.g., .