Graphs and Combinatorics

, Volume 32, Issue 6, pp 2339–2353 | Cite as

Some Triangulated Surfaces without Balanced Splitting

Original Paper
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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.

Keywords

Combinatorial surfaces Splitting cycles Complete graphs 

Mathematics Subject Classification

05C10 57M07 68R99 

Notes

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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Copyright information

© Springer Japan 2016

Authors and Affiliations

  1. 1.Gipsa-labUniversité de Grenoble AlpesGrenobleFrance
  2. 2.Gipsa-labCNRSGrenobleFrance

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