Face 2-colourable triangular embeddings of complete graphs

A face 2-colourable triangulation of an orientable surface by a complete graph K-n exists if and only if n = 3 or 7 (mod 12). The existence of such triangulations follows from current graph constructions used in the proof of the Heawood conjecture. In this paper we give an alternative construction for half of the residue class n=7 (mod 12) which lifts a face 2-colourable triangulation by K-m to one by K3m-2. A nonorientable version of this result is discussed as well which enables us to produce nonisomorphic nonorientable triangular embeddings of K-n for half of the residue class n = 1 (mod 6). We also note the existence of nonisomorphic orientable triangular embeddings of K-n for n = 7 (mod 12) and n not equal 7. (C) 1998 Academic Press.