Beyond Brooks: $(\Delta-1)$-Coloring in Semi-Streaming

Reed [J.~Comb.~Theory B, 1999] showed that graphs of maximum degree $\Delta \geq 10^{14}$ without $\Delta$-cliques are $(\Delta-1)$-colorable. We design a one-pass semi-streaming algorithm for computing such a coloring. Additionally, we prove that any one-pass $(\Delta-k)$-coloring algorithm for $0\leq k < (\Delta+1)/2$ requires $\Omega(n(k+1))$ space.