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 

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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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