Neighborhood graphs and distributed Δ+1-coloring
A tantalizing open question in the theory of distributed computing asks whether a graph with maximum degree Δ can be colored with Δ+1 colors in polylog deterministic steps in the distributed model of computation. Linial introduced the notion of a t-neighborhood graph of a given graph G and showed that the chromatic number of this graph is a lower bound on the number of colors that G can be colored with in t steps of the distributed model. In this paper we show that the chromatic number of any t-neighborhood graph is at most Δ + 1 for some t = O(log3n). This implies that current techniques for proving lower bounds on the distributed complexity of Δ + 1-coloring are not strong enough to give a negative answer to the above open problem. The proof of this result is based on the analysis of a randomized algorithm for this problem using martingale inequalities. We also show that in a nonconstructive sense the Δ+1-coloring problem can be solved in polylog time for an infinite class of graphs including vertex-transitive graphs.
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