Odd complete minors in even embeddings on surfaces

In this paper, we study the odd  KmKm-minor problem in even embeddings on surfaces. We first establish a general theory for even embeddings with odd KmKm-minors. Given an integer mm we show that for every surface F2F2 of sufficiently high genus there exists a constant N=N(F2)N=N(F2) so that every non-bipartite even embedding on F2F2 with representativity at least NN contains an odd KmKm as a minor. In the second part we prove that every 19-representative non-bipartite even embedding in an arbitrary orientable surface of genus ≥1≥1 has an odd K5K5-minor.