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Some Triangulated Surfaces without Balanced Splitting

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Abstract

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.

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Notes

  1. Two closed curves are homotopic if one can be continuously deformed into the other.

  2. In the non-orientable case, the left and right orientation should be propagated along the path \((v_0,\dots ,v_{k+1})\).

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Acknowledgments

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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Authors and Affiliations

  1. Gipsa-lab, Université de Grenoble Alpes, Grenoble, France

    Vincent Despré

  2. Gipsa-lab, CNRS, Grenoble, France

    Francis Lazarus

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  1. Vincent Despré
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  2. Francis Lazarus
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Correspondence to Francis Lazarus.

Additional information

This work has been partially supported by the LabEx PERSYVAL-Lab (ANR-11-LABX-0025-01).

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Despré, V., Lazarus, F. Some Triangulated Surfaces without Balanced Splitting. Graphs and Combinatorics 32, 2339–2353 (2016). https://doi.org/10.1007/s00373-016-1735-6

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  • DOI: https://doi.org/10.1007/s00373-016-1735-6

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