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    Some Triangulated Surfaces without Balanced Splitting

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    Let G be the graph of a triangulated surface Σ\Sigma of genus g≥2g\geq 2. A cycle of G is splitting if it cuts Σ\Sigma 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.Comment: 15 pages, 7 figure
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