Recursive constructions and nonisomorphic minimal nonorientable embeddings of complete graphs

We construct a family of recursive constructions such that for any i∈{0,1,3,4,6,7,9,10} and j∈{0,1,…,11}, several arbitrary nonorientable triangular embeddings of every complete graph Km, m≡i(mod12), can be incorporated into a minimal nonorientable embedding of Km̄, m̄≡j(mod12). The existence of such recursive constructions implies the following important interdependency of the sets of nonisomorphic minimal nonorientable embeddings of Kn for different residue classes of n modulo 12: if for some i∈{0,1,3,4,6,7,9,10}, the number of nonisomorphic nonorientable triangular embeddings of a graph Km, m≡i(mod12), is large enough, then for any other j∈{0,1,…,11}, the number of nonisomorphic minimal nonorientable embeddings of some graph Km̄, m̄≡j(mod12), is also large enough. As a consequence, using Grannell and Knor's (2013) results for Kn, n≡1 or 9(mod12), we show that there is a certain positive constant a such that for any i∈{0,1,…,11}, there is an infinite set (a linear class) of values of n, where n≡i(mod12), such that the number of nonisomorphic minimal nonorientable embeddings of Kn is at least nan2−o(n2) as n→∞.