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 (FP-TAS/FPRAS) for counting the number of proper colorings as long as q ≥ ∆ + 1, where q is the number of colors and ∆ is the maximum degree of the graph. The bound of q = ∆ + 1 is the uniqueness threshold for Gibbs measure on ∆-regular infinite trees. However, the conjecture remained open even for any fixed ∆ ≥ 3 (The cases of ∆ = 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 ∆ = 3. This is the first time to achieve this optimal bound of q = ∆ + 1. Previously, the best FPRAS requires q > 11 6 ∆ and the best deterministic FPTAS requires q > 2.581∆ + 1 for general graphs. In the case of ∆ = 3, the best previous result is an FPRAS for counting proper 5-colorings. We note that there is a barrier to go beyond q = ∆ + 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.
For the special case Δ = 3, Lu et al. REF give a correlation decay algorithm for counting 4-colorings.
An FPTAS for Counting Proper Four-Colorings on Cubic Graphs
ArXiv
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Mathematics Computer Science