On the convergence of the mild random walk algorithm to generate random one-factorizations of complete graphs

Abstract

The complete graph K n , for n even, has a one-factorization (proper edge coloring) with n – 1 colors. In the recent contribution [Dotan M., Linial N. (2017). ArXiv:1707.00477v2], the authors raised a conjecture on the convergence of the mild random walk on the Markov chain whose nodes are the colorings of K n . The mild random walk consists in moving from a coloring C to a recoloring C′ if and only if ϕ(C′) ≤ ϕ(C), where ϕ is the potential function that takes its minimum at one-factorizations. We show the validity of such algorithm with several numerical experiments that demonstrate convergence in all cases (not just asymptotically) with polynomial cost. We prove several results on the mild random walk, we study deeply the properties of local minimum colorings, we give a detailed proof of the convergence of the algorithm for K 4 and K 6, and we raise new conjectures. We also present an alternative to the potential measure ϕ by consider the Shannon entropy, which has a strong parallelism with ϕ from the numerical standpoint.