It is proved that a complete graph Kn can have an orientation whose minimum directed genus is ⌈1⁄12(n - 3)(n - 4)⌉ if and only if n ≡ 3, 7 (mod 12). This answers a question of Bonnington et al. by using a method different from current graphs. It is also proved that a complete symmetric tripartite graph Kn, n, n has an orientation whose minimum directed genus is 1⁄2(n - 1)(n - 2).
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China