The nonorientable genus of complete tripartite graphs

Abstract

In 1976, Stahl and White conjectured that the nonorientable genus of $K_{l,m,n}$, where $l⩾m⩾n$, is $\lceil (l-2)(m+n-2)/2\rceil$. 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.