Fewer colors for perfect simulation of proper colorings
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Given a graph G and color set {1, …, k}, a 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 > 3Δ, where Δ 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(Δ - 1) for all Δ≥ 2.
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