Sampling colorings almost uniformly in sparse random graphs

The problem of sampling proper q-colorings from uniform distribution has been extensively studied. Most of existing samplers require q ≥ α∆+ β for some constants α and β, where ∆ is the maximum degree of the graph. The problem becomes more challenging when the underlying graph has unbounded degree since even the decision of q-colorability becomes nontrivial in this situation. The Erdős-Rényi random graph G(n, d/n) is a typical class of such graphs and has received a lot of recent attention. In this case, the performance of a sampler is usually measured by the relation between q and the average degree d. We are interested in the fully polynomialtime almost uniform sampler (FPAUS) and the state-of-the-art with such sampler for proper q-coloring on G(n, d/n) requires that q ≥ 5.5d. In this paper, we design an FPAUS for proper q-colorings on G(n, d/n) by requiring that q ≥ 3d + O(1), which improves the best bound for the problem so far. Our sampler is based on the spatial mixing property of q-coloring on random graphs. The core of the sampler is a deterministic algorithm to estimate the marginal probability on blocks, which is computed by a novel block version of recursion for q-coloring on unbounded degree graphs.