The nonorientable genus of complete tripartite graphs

In 1976, Stahl and White conjectured that the nonorientable genus of K l , m , n , where l ⩾ m ⩾ n , is ⌈ ( l − 2 ) ( m + n − 2 ) / 2 ⌉ . The authors recently showed that the graphs K 3 , 3 , 3 , K 4 , 4 , 1 , and K 4 , 4 , 3 are counterexamples to this conjecture. Here we prove that apart from these three exceptions, the conjecture is true. In the course of the paper we introduce a construction called a transition graph, which is closely related to voltage graphs.