Some Triangulated Surfaces without Balanced Splitting

Let G be the graph of a triangulated surface Σ of genus g ≥ 2. A cycle of G is splitting if it cuts Σ into two components, neither of which is homeomorphic to a disk. A splitting cycle has type k if the corresponding components have genera k and g− k. It was conjectured that G contains a splitting cycle (Barnette ’1982). We confirm this conjecture for an infinite family of triangulations by complete graphs but give counter-examples to a stronger conjecture (Mohar and Thomassen ’2001) claiming that G should contain splitting cycles of every possible type.