# Random Bichromatic Matchings

- First Online:

- Received:
- Accepted:

DOI: 10.1007/s00453-007-9096-4

- Cite this article as:
- Bhatnagar, N., Randall, D., Vazirani, V.V. et al. Algorithmica (2008) 50: 418. doi:10.1007/s00453-007-9096-4

## Abstract

Given a graph with edges colored Red and Blue, we study the problem of sampling and approximately counting the number of matchings with exactly *k*Red edges. We solve the problem of estimating the number of perfect matchings with exactly *k*Red edges for dense graphs. We study a Markov chain on the space of all matchings of a graph that favors matchings with *k*Red edges. We show that it is rapidly mixing using non-traditional canonical paths that can backtrack. We show that this chain can be used to sample matchings in the 2-dimensional toroidal lattice of any fixed size *ℓ* with *k*Red edges, where the horizontal edges are Red and the vertical edges are Blue.