We present an efficient and scalable Coarse Grained Multicomputer (CGM) coloring algorithm that colors a graph G with at most D+ 1 colors where D is the maximum degree in G. This algorithm is given in two variants: randomized and deterministic. We show that on a p-processor CGM model the proposed algorithms require a parallel time of O( |G| p ) and a total work and overall communication cost of O(|G|). These bounds correspond to the average case for the randomized version and to the worst-case for the deterministic variant. Key words: graph algorithms, parallel algorithms, graph coloring, Coarse Grained Multicomputers 1

We consider the problem of edge-coloring planar graphs. It is known that a planar graph G with maximum degree \Delta \geq 8 can be colored with \Delta colors. We present two algorithms which find such a coloring when \Delta \geq 9. The first one is a sequential O(n log n) time algorithm. The other one is a parallel EREW PRAM algorithm which works in time O(log³ n) and uses O(n) processors.