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 in Graphs on surfaces. Johns Hopkins studies in the mathematical sciences. Johns Hopkins University Press, Baltimore, 2001) claiming that G should contain splitting cycles of every possible type.