The orientable genus of some joins of complete graphs with large edgeless graphs

In an earlier paper the authors showed that with one exception the nonorientable genus of the graph (K-m) over bar, + K-n with m >= n - 1, the join of a complete graph with a large edgeless graph, is the same as the nonorientable genus of the spanning subgraph (K-m) over bar + (K-n) over bar = K-m,K-n. The orientable genus problem for (K-m) over bar + K-n with m >= n - 1 seems to be more difficult, but ill this paper we find the orientable genus of some of these graphs. In particular, we determine the genus of (K-m) over bar + K-n when n is even and m >= n, the genus of (K-m) over bar + K-n when n = 2(p) + 2 for p >= 3 and m >= n - 1, and the genus of (K-m) over bar + K-n when n = 2(p) + 1 for p >= 3 and m >= n + 1. In all of these cases the genus is the same as the genus of K-m,K-n namely inverted right perpendicular(m - 2)(n - 2)/4inverted left perpendicular. (C) 2008 Elsevier B.V. All rights reserved.