Abstract. Face two-colourable embeddings of complete graphs correspond to biembeddings of Steiner triple systems. Such embeddings exist only if n is congruent to 1 or 3 modulo 6. In this paper we present the number of these embeddings for n=13 found by a computer programm.

There are two Steiner triple systems of order 13. One of them is cyclic (denote it by C) and the other is obtained from the cyclic system by so-called "Pasch switch" (denote it by N). We found that there are

  1. 615 nonisomorphic embeddings of C with C,
  2. 8,539 nonisomorphic embeddings of C with N,
  3. 29,454 nonisomorphic embeddings of N with N.
Summing up, there are 38,608 face two-colourable triangulations of K13.