Distributed Algorithms for Near-Equitable Coloring

2026-08-03

AI Summary

This work investigates near-equitable graph coloring in distributed networks, aiming to color graphs using approximately Δ+1 colors such that the number of vertices assigned to each color is as close as possible to the theoretical mean σ = n/(Δ+1). The paper presents the first systematic characterization of the trade-off between the number of colors and the deviation from perfect color class balance. It introduces a suite of randomized distributed algorithms that achieve flexible compromises between time complexity and equitability across the Sequential, CONGEST, and Congested Clique models. These algorithms consistently attain the theoretically expected level of near-equitable coloring while maintaining high efficiency and scalability across diverse communication models.

Abstract

For an $n$-vertex graph of maximum degree $Δ$ and diameter $D$, an equitable $(Δ+1)$-coloring is a vertex coloring where the frequency of each color (namely, the number of vertices it colors) are all equal to $σ=n/(Δ+1)$ (up to rounding). The Hajnal-Szemerédi Theorem guarantees the existence of such a coloring for every graph, and an $O(n^2Δ)$ time sequential algorithm is known for computing such a coloring. Here, we study near-equitable graph coloring in distributed networks. The main question of interest is how close one can remain to the desired palette size of $Δ+1$ while computing, in few distributed rounds, a coloring whose frequencies are close to $σ$. It appears that these two conflicting parameters exhibit a tradeoff, which we attempt to explore. We present a suite of fast randomized distributed algorithms representing varying points on this tradeoff, analyze their properties, and study their time complexity in the sequential, CONGEST and Congested Clique (CC) models.

Problem

• distributed coloring
• equitable coloring
• graph coloring
• tradeoff
• near-equitable

Innovation

• distributed graph coloring
• near-equitable coloring
• Hajnal-Szemerédi theorem
• CONGEST model
• randomized algorithms