k-Coloring is Faster than Computing the Chromatic Number

Author: Or Zamir
arXiv Identifier: 2607.25973 (Category: cs.DS)
Submission Date: July 28, 2026

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📌 Executive Summary

This research demonstrates that the problem of $k$-coloring for graphs with $n$ vertices can be solved using a randomized algorithm with a time complexity of $(2-\varepsilon_k)^n$, provided that $\varepsilon_k > 0$ for any fixed value of $k$.

The Core Breakthrough

For a long time, the benchmark for computing the chromatic number was the $O^\star\bigl(2^n\bigr)$ algorithm established by Björklund, Husfeldt, and Koivisto in 2009. Until now, specialized algorithms that outperformed this general bound were only available for cases where $\text{~~}k \leq 6\text{~~}$. This paper resolves that gap, proving that $k$-coloring is fundamentally faster than calculating the general chromatic number for any fixed $k$.

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🛠️ Methodology & Technical Approach

The author achieves this result by synthesizing and expanding upon several advanced theoretical frameworks. The process involves a sophisticated chain of reductions:

1. Theoretical Foundations

The solution integrates tools from two previous works by Zamir:

• ICALP 2021: A reduction method moving from $(k+2)$-coloring to $k$-list-coloring.
• STOC 2023: An approach utilizing hypergraph-containers.

2. New Contributions

To bridge the remaining gaps, the paper introduces:

• Novel algorithms designed for list-coloring instances that feature a mixture of "short" and "long" color lists.
• An iterable reduction mechanism that transforms $(k+1)$-list-coloring problems into $k$-list-coloring problems using fixed palettes.

3. Logic Flow

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📊 Comparison of Complexity

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📝 Implementation Details

The complexity can be viewed as a function of the vertex count $n$:

def time_complexity(n, epsilon_k):
    # The new randomized bound
    return (2 - epsilon_k)**n

Key Objectives Achieved:

• Generalize $(k+2)$-coloring reductions.
• Incorporate hypergraph-container theory.
• Develop algorithms for mixed-length color lists.
• Establish the iterable reduction for fixed palettes.

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🌐 Metadata & Archive Information

Subjects: Data Structures and Algorithms (cs.DS)
DOI: https://doi.org/10.48550/arXiv.2607.25973

Submission History

• v1: Submitted Tuesday, 28 Jul 2026, 16:53:58 UTC (Size: 47 KB)

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• Source: [TeX Source]

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