Improved bounds for perfect sampling of k-colorings in graphs
Siddharth Bhandari, Sayantan Chakraborty · 2020
We present a randomized algorithm that takes as input an undirected n-vertex graph G with maximum degree Δ and an integer k > 3Δ, and returns a random proper k-coloring of G. The distribution of the coloring is perfectly uniform over the set of all proper k-colorings; the expected running time of the algorithm is poly(k,n)=O(nΔ2· log(k)). This improves upon a result of Huber (STOC 1998) who obtained a polynomial time perfect sampling algorithm for k>Δ2+2Δ. Prior to our work, no algorithm with expected running time poly(k,n) was known to guarantee perfectly sampling with sub-quadratic number of colors in general.