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

M. J. Grannell1, T. S. Griggs1, M. Knor2
1Department of Pure Mathematics The Open University Walton Hall Milton Keynes MK7 6AA UNITED KINGDOM
2Department of Mathematics Faculty of Civil Engineering Slovak University of Technology Radlinského 11 813 68 Bratislava SLOVAKIA

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 \( n = 13 \).