The non-orientable genus of some metacyclic groups

In the case of a 2-cell embedding (i.e., in the case where every region of the embedding is homeomorphic to a disk), the'inequality can easily be derived from the Euler Formula, v e + f = 2 2"fwhere e is the number of edges and f is the number of 2-cells of the embedding): the Handshaking Lemma asserts 2e = dv, and applying the Handshaking Lemma to the dual graph yields 2e >. gf; incorporating these observations into the Euler Formula and using the fact that 7 is an integer results in precisely (1.1). For an argument extending this inequality to the case where not every region is homeomorphic to a disk, see [12]. A similar result is true for an embedding into a non-orientable surface of nonorientable genus y [8, Theorem 2b]. In this case, we use the non-orientable version of the Euler Formula, v e + f = 2 +, and the conclusion is