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

TL;DR

This paper introduces a semi-streaming algorithm for $( ext{max degree}-1)$-coloring of graphs without large cliques and establishes space complexity lower bounds for related coloring algorithms.

Contribution

It presents the first one-pass semi-streaming algorithm for $( ext{max degree}-1)$-coloring in certain graphs and proves space lower bounds for $( ext{max degree}-k)$-coloring algorithms.

Findings

Developed a semi-streaming algorithm for $( ext{max degree}-1)$-coloring.

Proved space complexity lower bounds for one-pass $( ext{max degree}-k)$-coloring algorithms.

Extended understanding of coloring in the semi-streaming model.

Abstract

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.