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

2026-05-08

AI Summary

This work addresses the (Δ−1)-coloring problem for graphs with maximum degree Δ ≥ 10¹⁴ that contain no Δ-clique, studied in the semi-streaming model. We present the first single-pass semi-streaming algorithm that achieves a (Δ−1)-coloring as guaranteed by Reed’s Theorem, marking the first such result in this model. Furthermore, we establish a space complexity lower bound of Ω(n(k+1)) for any single-pass (Δ−k)-coloring algorithm. By constructing an efficient coloring algorithm under the stated conditions and simultaneously proving a matching theoretical lower bound on space requirements, this study bridges a critical gap at the intersection of graph coloring and streaming algorithms, advancing both the algorithmic and complexity-theoretic understanding of coloring problems in the semi-streaming setting.

Abstract

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

Problem

• graph coloring
• semi-streaming
• maximum degree
• clique
• space complexity

Innovation

• semi-streaming
• graph coloring
• one-pass algorithm
• space complexity
• Brooks' theorem