Face two-Colourable Triangulations of \( K_{13} \)

Abstract

Face two-colourable triangular embeddings of complete graphs \(K_n\) 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 \( n = 13 \).