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Distance-2 Coloring in the CONGEST Model

We give efficient randomized and deterministic distributed algorithms for computing a distance-$2$ vertex coloring of a graph $G$ in the CONGEST model. In particular, if $Δ$ is the maximum degree of $G$, we show that there is a randomized CONGEST model algorithm to compute a distance-$2$ coloring of $G$ with $Δ^2+1$ colors in $O(\logΔ\cdot\log n)$ rounds. Further if the number of colors is slightly increased to $(1+ε)Δ^2$ for some $ε>1/{\rm polylog}(n)$, we show that it is even possible to compute a distance-$2$ coloring deterministically in polylog$(n)$ time in the CONGEST model. Finally, we give a $O(Δ^2 + \log^* n)$-round deterministic CONGEST algorithm to compute distance-$2$ coloring with $Δ^2+1$ colors.