Efficient, local and symmetric Markov chains that generate one-factorizations

Abstract

It is well known that for every even integer n, the complete graph $K_{n}$ has a one-factorization, namely a proper edge coloring with $n-1$ colors. Unfortunately, not much is known about the possible structure of large one-factorizations. Also, at present we have only woefully few explicit constructions of large one-factorizations. In particular, we know essentially nothing about the typical properties of one-factorizations for large n. In this view, it is desirable to find rapidly mixing Markov chains that generate one-factorizations uniformly and efficiently. No such Markov chain is currently known, and here we take a step this direction. We construct a Markov chain whose states are all edge colorings of $K_n$ by $n-1$ colors. This chain is invariant under arbitrary renaming of the vertices or the colors. It reaches a one factorization in polynomial(n) steps from every starting state. In addition, at every transition only O(n) edges change their color. We also raise some related questions and conjectures, and present results of numerical simulations of simpler variants of this Markov chain.