Counterexamples to the nonorientable genus conjecture for complete tripartite graphs

In 1976, Stahl and White conjectured that the minimum nonorientable genus of Kl,m,n (where l≥m≥n) is (l−2)(m+n−2)2. We prove that K4,4,1,K4,4,3, and K3,3,3 are counterexamples to this conjecture. We also show that all other complete tripartite graphs Kl,m,n with l≥m≥n and l≤5 satisfy the conjecture. Moreover, all complete tripartite graphs with l≤5 satisfy the similar conjecture for orientable genus.