Some Triangulated Surfaces without Balanced Splitting
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Let G be the graph of a triangulated surface \(\varSigma \) of genus \(g\ge 2\). A cycle of G is splitting if it cuts \(\varSigma \) 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.
KeywordsCombinatorial surfaces Splitting cycles Complete graphs
Mathematics Subject Classification05C10 57M07 68R99
The authors would like to thank Thom Sulanke for interesting discussions about his code for generating irreducible triangulations. We are also grateful to the anonymous reviewer for her/his detailed comments and suggestions.
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