# Linear-Time and Efficient Distributed Algorithms for List Coloring Graphs on Surfaces

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 such 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. Dvorak and Kawarabayashi actually gave an O(n^O(g+1))-time algorithm to find such colorings (if they exist) in n-vertex graphs where g is the Euler genus of the surface. Here we provide the first such algorithm whose exponent does not depend on the genus; indeed, we provide a linear-time algorithm. In 1988, Goldberg, Plotkin and Shannon provided a deterministic distributed algorithm for 7-coloring n-vertex planar graphs in O( n) rounds. In 2018, Aboulker, Bonamy, Bousquet, and Esperet provided a deterministic distributed algorithm for 6-coloring n-vertex planar graphs in O(^3 n) rounds. Their algorithm in fact works for 6-list-coloring. They also provided an O(^3 n)-round algorithm for 4-list-coloring triangle-free planar graphs. Chechik and Mukhtar independently obtained such algorithms for ordinary coloring in O( 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( n) rounds, work also for list-colorings, and even work on a fixed surface (assuming such a coloring exists).

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