Face 2-Colourable Triangular Embeddings of Complete Graphs

A face 2-colourable triangulation of an orientable surface by a complete graphKnexists if and only ifn≡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 classn≡7 (mod≡12) which lifts a face 2-colourable triangulation byKmto one byK3m−2. A nonorientable version of this result is discussed as well which enables us to produce nonisomorphic nonorientable triangular embeddings ofKnfor half of the residue classn≡1 (mod 6). We also note the existence of nonisomorphic orientable triangular embeddings ofKnforn≡7 (mod 12) andn≠7.