Fewer colors for perfect simulation of proper colorings

Abstract

Given a graph $G$ and color set $\{1, \ldots, k\}$, a $\textit{proper coloring}$ is an assignment of a color to each vertex of $G$ such that no two vertices connected by an edge are given the same color. The problem of drawing a proper coloring exactly uniformly from the set of proper colorings is well-studied. Most recently, Bhandari and Chakraborty developed a polynomial expected time randomized algorithm for obtaining such draws when $k > 3\Delta$, where $\Delta$ is the maximum degree of the graph. Their approach used a bounding chain together with the coupling from the past protocol. Here a new randomized algorithm is presented based upon the randomness recycler protocol introduced by the author and Fill at FOCS 2000. Given $n$ vertices, this method takes $O(n \ln (n))$ expected steps when $k > 2.27(\Delta - 1)$ for all $\Delta \geq 2$.