Some Triangulated Surfaces without Balanced Splitting

TL;DR

This paper investigates splitting cycles in triangulated surfaces, confirming their existence for an infinite family of graphs but providing counterexamples to a stronger conjecture about all possible cycle types.

Contribution

It proves the existence of splitting cycles in certain triangulations and refutes a broader conjecture about their types in all cases.

Findings

Confirmed conjecture for an infinite family of triangulations

Provided counterexamples to the stronger conjecture

Showed not all cycle types are necessarily present

Abstract

Let G be the graph of a triangulated surface $\Sigma$ of genus $g\geq 2$. A cycle of G is splitting if it cuts $\Sigma$ 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 '2001) claiming that G should contain splitting cycles of every possible type.