Optimal Mixing for Randomly Sampling Edge Colorings on Trees Down to the Max Degree

We address the convergence rate of Markov chains for randomly generating an edge coloring of a given tree. Our focus is on the Glauber dynamics which updates the color at a randomly chosen edge in each step. For a tree $T$ with $n$ vertices and maximum degree $Δ$, when the number of colors $q$ satisfies $q\geqΔ+2$ then we prove that the Glauber dynamics has an optimal relaxation time of $O(n)$, where the relaxation time is the inverse of the spectral gap. This is optimal in the range of $q$ in terms of $Δ$ as Dyer, Goldberg, and Jerrum (2006) showed that the relaxation time is $Ω(n^3)$ when $q=Δ+1$. For the case $q=Δ+1$, we show that an alternative Markov chain which updates a pair of neighboring edges has relaxation time $O(n)$. Moreover, for the $Δ$-regular complete tree we prove $O(n\log^2{n})$ mixing time bounds for the respective Markov chain. Our proofs establish approximate tensorization of variance via a novel inductive approach, where the base case is a tree of height $\ell=O(Δ^2\log^2Δ)$, which we analyze using a canonical paths argument.