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 $\Delta$ 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 $\Delta^2+1$ colors in $O(\log\Delta\cdot\log n)$ rounds. Further if the number of colors is slightly increased to $(1+\epsilon)\Delta^2$ for some $\epsilon>1/{\rm polylog}(n)$, 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 $O(\Delta^2 + \log^* n)$-round deterministic CONGEST algorithm to compute distance-$2$ coloring with $\Delta^2+1$ colors.