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 ¯ + 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 ¯ + K n ¯ = K m , n . The orientable genus problem for K m ¯ + K n with m ≥ n − 1 seems to be more difficult, but in this paper we find the orientable genus of some of these graphs. In particular, we determine the genus of K m ¯ + K n when n is even and m ≥ n , the genus of K m ¯ + K n when n = 2 p + 2 for p ≥ 3 and m ≥ n − 1 , and the genus of K m ¯ + 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 , n , namely ⌈ ( m − 2 ) ( n − 2 ) / 4 ⌉ .