Simple Graph Coloring Algorithms for Congested Clique and Massively Parallel Computation.

We present a very simple randomized partitioning procedure for graph coloring, which leads to simplification or improvements of some recent distributed and parallel coloring algorithms. In particular, we get a simple $(Delta+1)$ coloring algorithm with round complexity $O(log^* Delta)$ in the CONGESTED CLIQUE model of distributed computing. This matches the bound of Parter and Su [DISCu002718], which improved on the $O(loglog Delta log^* Delta)$-round algorithm of Parter [ICALPu002718]. Moreover, the same random partitioning leads to a $(Delta+1)$ coloring algorithm with round complexity $O(log^* Delta+ sqrt{loglog n})$ in the Massively Parallel Computation (MPC) model with strongly sublinear memory, which is the first sublogarithmic-time algorithm in this regime. This algorithm uses a memory of $O(n^{alpha})$ per machine, for any desirable constant $alphau003e0$, and a total memory of $widetilde{O}(m)$, where $m$ is the size of the graph.