In 1994, Thomassen proved that every planar graph is 5-list-colorable. In 1995, Thomassen proved that every planar graph of girth at least five is 3-list-colorable. His proofs naturally lead to quadratic-time algorithms to find such colorings. Here, we provide the first linear-time algorithms to find such colorings. For a fixed surface S, Thomassen showed in 1997 that there exists a linear-time algorithm to decide if a graph embedded in S is 5-colorable and similarly in 2003 if a graph of girth at least five embedded in S is 3-colorable. Using the theory of hyperbolic families, the author and Thomas showed such algorithms exist for list-colorings. Around the same time, Dvorak and Kawarabayashi also provided such algorithms. Moreover, they gave an algorithm to find such colorings (if they exist). Here we provide the first such algorithm which is fixed parameter tractable with genus as the parameter; indeed, we provide a linear-time algorithm to find such colorings. In 1988, Goldberg, Plotkin and Shannon provided a deterministic distributed algorithm for 7-coloring n-vertex planar graphs in O(log n) rounds. In 2018, Aboulker, Bonamy, Bousquet, and Esperet provided a deterministic distributed algorithm for 6-coloring n-vertex planar graphs in polylogarithmic rounds. Their algorithm in fact works for 6-list-coloring. They also provided a polylogarithmic algorithm for 4-list-coloring triangle-free planar graphs. Chechik and Mukhtar independently obtained such algorithms for ordinary coloring in O(log n) rounds, which is best possible in terms of running time. Here we provide the first polylogarithmic deterministic distributed algorithms for 5-coloring n-vertex planar graphs and similarly for 3-coloring planar graphs of girth at least five. Indeed, these algorithms run in O(log n) rounds, work also for list-colorings, and even work on a fixed surface (assuming such a coloring exists).

A well-known conjecture in computer science and statistical physics is that Glauber dynamics on the set of k-colorings of a graph G on n vertices with maximum degree Δ is rapidly mixing for k ≥ Δ + 2. In FOCS 1999, Vigoda [43] showed that the flip dynamics (and therefore also Glauber dynamics) is rapidly mixing for any . It turns out that there is a natural barrier at , below which there is no one-step coupling that is contractive with respect to the Hamming metric, even for the flip dynamics. We use linear programming and duality arguments to fully characterize the obstructions to going beyond . These extremal configurations turn out to be quite brittle, and in this paper we use this to give two proofs that the Glauber dynamics is rapidly mixing for any for some absolute constant ε0 > 0. This is the first improvement to Vigoda's result that holds for general graphs. Our first approach analyzes a variable-length coupling in which these configurations break apart with high probability before the coupling terminates, and our other approach analyzes a one-step path coupling with a new metric that counts the extremal configurations. Additionally, our results extend to list coloring, a widely studied generalization of coloring, where the previously best known results required k > 2Δ.