An FPTAS for Counting Proper Four-Colorings on Cubic Graphs

Graph coloring is arguably the most exhaustively studied problem in the area of approximate counting. It is conjectured that there is a fully polynomial-time (randomized) approximation scheme (FPTAS/FPRAS) for counting the number of proper colorings as long as q >= Delta + 1, where q is the number of colors and Delta is the maximum degree of the graph. The bound of q = Delta + 1 is the uniqueness threshold for Gibbs measure on Delta-regular infinite trees. However, the conjecture remained open even for any fixed Delta >= 3 (The cases of Delta = 1, 2 are trivial). In this paper, we design an FPTAS for counting the number of proper four-colorings on graphs with maximum degree three and thus confirm the conjecture in the case of Delta = 3. This is the first time to achieve this optimal bound of q = Delta + 1. Previously, the best FPRAS requires q > 11/6 Delta and the best deterministic FPTAS requires q > 2.581 Delta + 1 for general graphs. In the case of Delta = 3, the best previous result is an FPRAS for counting proper 5-colorings. We note that there is a barrier to go beyond q = Delta + 2 for single-site Glauber dynamics based FPRAS and we overcome this by correlation decay approach. Moreover, we develop a number of new techniques for the correlation decay approach which can find applications in other approximate counting problems.