# The local nature of Δ-coloring and its algorithmic applications

- Received:

DOI: 10.1007/BF01200759

- Cite this article as:
- Panconesi, A. & Srinivasan, A. Combinatorica (1995) 15: 255. doi:10.1007/BF01200759

## Abstract

Given a connected graph*G*=(*V, E*) with |*V*|=*n* and maximum degree Δ such that*G* is neither a complete graph nor an odd cycle, Brooks' theorem states that*G* can be colored with Δ colors. We generalize this as follows: let*G*-*v* be Δ-colored; then,*v* can be colored by considering the vertices in an*O*(log_{Δ}*n*) radius around*v* and by recoloring an*O*(log_{Δ}*n*) length “augmenting path” inside it. Using this, we show that Δ-coloring*G* is reducible in*O*(log^{3}*n*/logΔ) time to (Δ+1)-vertex coloring*G* in a distributed model of computation. This leads to fast distributed algorithms and a linear-processor*NC* algorithm for Δ-coloring.