# Balanced subdivisions and flips on surfaces

## TL;DR

This paper investigates the connectivity of balanced triangulations of surfaces, demonstrating that certain local operations can connect all such triangulations on a sphere except for octahedral cases.

## Contribution

It introduces new local operations and proves their sufficiency for connecting balanced triangulations of surfaces, answering a question by Izmestiev, Klee, and Novik.

## Key findings

- Balanced triangulations are not always connected by stellar subdivisions and welds.
- Three local operations suffice to connect any two balanced triangulations of a surface.
- On the 2-sphere, pentagon contractions connect all non-octahedral triangulations.

## Abstract

In this paper, we show that two balanced triangulations of a closed surface are not necessary connected by a sequence of balanced stellar subdivisions and welds. This answers a question posed by Izmestiev, Klee and Novik. We also show that two balanced triangulations of a closed surface are connected by a sequence of three local operations, which we call the pentagon contraction, the balanced edge subdivision and the balanced edge weld. In addition, we prove that two balanced triangulations of the 2-sphere are connected by a sequence of pentagon contractions and their inverses if none of them are octahedral spheres.