Distributed (Delta + 1)-coloring in linear (in Delta) time
  The distributed (Delta + 1)-coloring problem is one of most fundamental and well-studied problems of Distributed Algorithms. Starting with the work of Cole and Vishkin in 86, there was a long line of gradually improving algorithms published. The current state-of-the-art running time is O(Delta log Delta + log^* n), due to Kuhn and Wattenhofer, PODC'06. Linial (FOCS'87) has proved a lower bound of 1/2 \log^* n for the problem, and Szegedy and Vishwanathan (STOC'93) provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve running time smaller than \Theta(Delta log Delta).   We present a deterministic (Delta + 1)-coloring distributed algorithm with running time O(Delta) + 1/2 log^* n. We also present a tradeoff between the running time and the number of colors, and devise an O(Delta * t)-coloring algorithm with running time O(Delta / t + \log^* n), for any parameter t, 1 < t < Delta^{1-epsilon}, for an arbitrarily small constant epsilon, 0 < epsilon < 1.   On the way to this result we study a generalization of the notion of graph coloring, which is called defective coloring. In an m-defective p-coloring the vertices are colored with p colors so that each vertex has up to m neighbors with the same color. We show that an m-defective p-coloring with reasonably small m and p can be computed very efficiently. We also develop a technique to employ multiple defect colorings of various subgraphs of the original graph G for computing a (Delta+1)-coloring of G. We believe that these techniques are of independent interest. 
Leonid Barenboim and Michael Elkin