MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Is there a way to sample a planar map uniformly at random? I am aware of the Cori-Vauquelin-Schaeffer bijection that can be used to sample and study uniformly random quadrangulations. There are other results in the literature that allow for sampling other classes of planar maps, but I haven't seen any results for the entire class of planar maps.

share|cite|improve this question
up vote 1 down vote accepted

Something you may potentially be interested in :


share|cite|improve this answer

Planar quadrangulation with $n$ faces are in bijection with planar maps with $n$ edges. The quadrangulation is bi-partite, so colour its vertices white/black. inside each face, add a diagonal edge between the two black vertices. Then delete all white vertices and incident edges. The result is a map with $n$ edges, and it is not hard to see that this is a bijection.

In fact, this is one of the reasons quadrangulations are a particularly interesting class of planar maps.

share|cite|improve this answer
Are there methods for sampling fixing other statistics (e.g. number of vertices or number of faces)? – Zach Hamaker Sep 23 '12 at 17:56

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.