The local nature of Δ-coloring and its algorithmic applications

Given a connected graphG=(V, E) with |V|=n and maximum degree Δ such thatG is neither a complete graph nor an odd cycle, Brooks' theorem states thatG can be colored with Δ colors. We generalize this as follows: letG-v be Δ-colored; then,v can be colored by considering the vertices in anO(logΔn) radius aroundv and by recoloring anO(logΔn) length “augmenting path” inside it. Using this, we show that Δ-coloringG is reducible inO(log3n/logΔ) time to (Δ+1)-vertex coloringG in a distributed model of computation. This leads to fast distributed algorithms and a linear-processorNC algorithm for Δ-coloring.