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arXiv · 1509.00269

Some Triangulated Surfaces without Balanced Splitting

Vincent Despré·Francis Lazarus
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Abstract

Let G be the graph of a triangulated surface $Σ$ of genus $g\geq 2$. 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.

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Vincent Despré, Francis Lazarus. 2015-09-01. Some Triangulated Surfaces without Balanced Splitting. https://arxiv.org/abs/1509.00269

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