We present a simple deterministic distributed algorithm that computes a $(Delta+1)$-vertex coloring in $O(log^2 Delta cdot log n)$ rounds. The algorithm can be implemented with $O(log n)$-bit messages. The algorithm can also be extended to the more general $(degree+1)$-list coloring problem. Obtaining a polylogarithmic-time deterministic algorithm for $(Delta+1)$-vertex coloring had remained a central open question in the area of distributed graph algorithms since the 1980s, until a recent network decomposition algorithm of Rozhov{n} and Ghaffari [STOC20]. The current state of the art is based on an improved variant of their decomposition, which leads to an $O(log^5 n)$-round algorithm for $(Delta+1)$-vertex coloring. Our coloring algorithm is completely different and considerably simpler and faster. It solves the coloring problem in a direct way, without using network decomposition, by gradually rounding a certain fractional color assignment until reaching an integral color assignments. Moreover, via the approach of Chang, Li, and Pettie [STOC18], this improved deterministic algorithm also leads to an improvement in the complexity of randomized algorithms for $(Delta+1)$-coloring, now reaching the bound of $O(log^3log n)$ rounds. As a further application, we also provide faster deterministic distributed algorithms for the following variants of the vertex coloring problem. In graphs of arboricity $a$, we show that a $(2+epsilon)a$-vertex coloring can be computed in $O(log^3 acdotlog n)$ rounds. We also show that for $Deltageq 3$, a $Delta$-coloring of a $Delta$-colorable graph $G$ can be computed in $O(log^2 Deltacdotlog^2 n)$ rounds.
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