Abstract

The technique of using di#erential equations to approximate the mean path of Markov chains has proved very useful in the average-case analysis of algorithms. Here, we significantly expand the range of this technique, by showing that it can be used to handle algorithms that favor high-degree vertices. In particular, we consider the problem of 3-coloring sparse random graphs and analyze a "smoothed" version of the Brelaz heuristic. This allows us to prove that i) almost all graphs with average degree d, i.e. G(n, p = d/n), are 3-colorable for d 4.03, and that ii) almost all 4-regular graphs are 3-colorable. This improves over the previous lower bound of 3.847 for the G(n, p) 3-colorability threshold and gives the first non-trivial result on the 3-colorability of random regular graphs.

Citations