Biembedding a Steiner Triple System With a Hamilton Cycle Decomposition of a Complete Graph

We construct a face two-colourable, blue and green say, embedding of the complete graph K-n in a nonorientable surface in which there are (n - 1)/2 blue faces each of which have a hamilton cycle as their facial walk and n(n - 1)/6 green faces each of which have a triangle as their facial walk; equivalently a biembedding of a Steiner triple system of order n with a hamilton cycle decomposition of K-n, for all n equivalent to 3 (mod 36) and n not equal 3. Using a variant of this construction, we establish the minimum genus of nonorientable embeddings of the graph K36k+3 + (K-m) over bar, for m = 18k + 1 + 6s where k >= 1 and 0 <= s <= k - 1. (C) 2013 Wiley Periodicals, Inc.