A note on the directed genus of Kn, n, n and Kn

It is proved that a complete graph K n 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 K n ,  n ,  n has an orientation whose minimum directed genus is 1 ⁄ 2 ( n - 1)( n - 2) .

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