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 $q \geq Δ+ 1$, where $q$ is the number of colors and $Δ$ is the maximum degree of the graph. The bound of $q = Δ+ 1$ is the uniqueness threshold for Gibbs measure on $Δ$-regular infinite trees. However, the conjecture remained open even for any fixed $Δ\geq 3$ (The cases of $Δ=1, 2$ are trivial). In this paper, we design an FPTAS for counting the number of proper $4$-colorings on graphs with maximum degree $3$ and thus confirm the conjecture in the case of $Δ=3$. This is the first time to achieve this optimal bound of $q = Δ+ 1$. Previously, the best FPRAS requires $q > \frac{11}{6} Δ$ and the best deterministic FPTAS requires $q > 2.581Δ+ 1$ for general graphs. In the case of $Δ=3$, the best previous result is an FPRAS for counting proper 5-colorings. We note that there is a barrier to go beyond $q = Δ+ 2$ 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.