Some Triangulated Surfaces without Balanced Splitting

Vincent Despré (1, 2) , Francis Lazarus (1)

1 GIPSA-AGPIG - GIPSA - Architecture, Géométrie, Perception, Images, Gestes
2 G-SCOP_OC - Optimisation Combinatoire

Let G be the graph of a triangulated surface Σ of genus g≥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 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.

Domaines: Géométrie algorithmique [cs.CG]; Mathématique discrète [cs.DM]

Vincent Despré, Francis Lazarus. Some Triangulated Surfaces without Balanced Splitting. Graphs and Combinatorics, 2016, 32 (6), pp.2339-2353. ⟨10.1007/s00373-016-1735-6⟩. ⟨hal-01705333⟩