On Approximately Counting Colorings of Small Degree Graphs

We consider approximate counting of colorings of an n-vertex graph using rapidly mixing Markov chains. It has been shown by Jerrum and by Salas and Sokal that a simple random walk on graph colorings would mix rapidly, provided the number of colors k exceeded the maximum degree ∆ of the graph by a factor of at least 2. We prove that this is not a necessary condition for rapid mixing by considering the simplest case of 5-coloring graphs of maximum degree 3. Our proof involves a computer-assisted proof technique to establish rapid mixing of a new “heat bath” Markov chain on colorings using the method of path coupling. We outline an extension to 7-colorings of triangle-free 4-regular graphs. Since rapid mixing implies approximate counting in polynomial time, we show in contrast that exact counting is unlikely to be possible (in polynomial time). We give a general proof that the problem of exactly counting the number of proper k-colorings of graphs with maximum degree ∆ is #P -complete whenever k ≥ 3 and ∆ ≥ 3.