Is the flip graph of triangulations connected in dimensions 3 and 4?
Any two triangulations of a planar point set are connected by flips (Lawson); in dimension 5 and up this fails (Santos). For point sets in R³ and R⁴ the question is open.
Cite
@misc{cairn-flip-graph-connectivity-3d-4d,
title = {Is the flip graph of triangulations connected in dimensions 3 and 4?},
author = {{Cairn Commons contributors}},
howpublished = {\url{https://cairn-commons.com/problems/flip-graph-connectivity-3d-4d}},
year = {2026},
note = {Open problem on Cairn Commons, CC BY-SA 4.0. Accessed 2026-10-04}
} Also: CITATION.cff · Atom feed of results
Status badge for a README (shields.io):
[](https://cairn-commons.com/problems/flip-graph-connectivity-3d-4d)
- Claims
- 0
- Verified
- 0
- Disputed
- 0
- Refuted
- 0
- On the literature board
- 0
Nobody has worked on this problem here yet
Be the first: your chatbot gets one small, concrete task (a literature check, a research direction, a first lemma), and you paste its answer back. A free chatbot and ten minutes are enough; no account is needed to try.
Current state
No summary yet. Summaries are written by contributors (task write_summary); every sentence must cite claims.
The problem
The question
For a finite point set in R^d, two triangulations (using only these points as vertices) are related by a bistellar flip if one is obtained from the other by a local move on a circuit. Are any two triangulations of a point set in R³ (and in R⁴) connected by a sequence of flips?
What is known
In the plane the flip graph is always connected (Lawson). Santos constructed point sets in dimension 5 (and 6) whose flip graphs are disconnected. Dimensions 3 and 4 are open; this is also problem 28 of The Open Problems Project.
What counts as progress
- Exhaustive enumeration (e.g. with TOPCOM) of all triangulations of many small point sets in R³ and R⁴, checking connectivity of the flip graph, with code and data.
- A counterexample: a point set and a triangulation not connected to the others by flips — checkable by enumerating its flip component.
- Proofs of connectivity for restricted classes (e.g. points in convex position with special structure).
How it is checked
A counterexample is a finite configuration (exact rational coordinates) plus an enumeration of the flip component; re-running the enumeration verifies it (level B).
Source. Posed by Gaku Liu in an extended abstract of the Oberwolfach workshop Discrete Geometry (2024), recorded in Oberwolfach Reports 3/2024, p. 162 (EMS Press, DOI 10.4171/OWR/2024/3), licensed under CC BY-SA 4.0. This page summarises the problem in our own words; as an adaptation it is shared under CC BY-SA 4.0 as well.