---
title: Simultaneous Diagonal Flips in Plane Triangulations
url: https://www.emergentmind.com/papers/0509478
type: paper
arxiv_id: '0509478'
arxiv_url: https://arxiv.org/abs/0509478
published: '2005-09-21'
categories:
- math.CO
- cs.CG
---

# Simultaneous Diagonal Flips in Plane Triangulations

## Abstract

Simultaneous diagonal flips in plane triangulations are investigated. It is proved that every $n$-vertex triangulation with at least six vertices has a simultaneous flip into a 4-connected triangulation, and that it can be computed in O(n) time. It follows that every triangulation has a simultaneous flip into a Hamiltonian triangulation. This result is used to prove that for any two $n$-vertex triangulations, there exists a sequence of $O(\log n)$ simultaneous flips to transform one into the other. The total number of edges flipped in this sequence is O(n). The maximum size of a simultaneous flip is then studied. It is proved that every triangulation has a simultaneous flip of at least ${1/3}(n-2)$ edges. On the other hand, every simultaneous flip has at most $n-2$ edges, and there exist triangulations with a maximum simultaneous flip of ${6/7}(n-2)$ edges.