[2607.25973] k-Coloring is Faster than Computing the Chromatic Number

0$ for every fixed $k$. Previously, only the cases $k\leq 6$ were known to have faster solutions than the general $O^\star\bigl(2^n\bigr)$ time algorithm of [Björklund, Husfeldt, Koivisto, SICOMP 2009] that computes the chromatic number.<br>We resolve this long-standing open problem by generalizing and combining tools from the $(k+2)$-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023]. Together with new algorithms for list-coloring instances mixing long and short color lists, this yields an iterable reduction from $(k+1)$-list-coloring to $k$-list-coloring over fixed palettes."/>

0$ for every fixed $k$. Previously, only the cases $k\leq 6$..."/>

0$ for every fixed $k$. Previously, only the cases $k\leq 6$ were known to have faster solutions than the general $O^\star\bigl(2^n\bigr)$ time algorithm of [Bj\"{o}rklund, Husfeldt, Koivisto, SICOMP 2009] that computes the chromatic number. We resolve this long-standing open problem by generalizing and combining tools from the $(k+2)$-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023]. Together with new algorithms for list-coloring instances mixing long and short color lists, this yields an iterable reduction from $(k+1)$-list-coloring to $k$-list-coloring over fixed palettes." />

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Computer Science > Data Structures and Algorithms

arXiv:2607.25973 (cs)

[Submitted on 28 Jul 2026]

Title:k-Coloring is Faster than Computing the Chromatic Number

Authors:Or Zamir<br>View a PDF of the paper titled k-Coloring is Faster than Computing the Chromatic Number, by Or Zamir

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Abstract:We prove that $k$-coloring on $n$-vertex graphs has a randomized algorithm running in time $(2-\varepsilon_k)^n$, where $\varepsilon_k>0$ for every fixed $k$. Previously, only the cases $k\leq 6$ were known to have faster solutions than the general $O^\star\bigl(2^n\bigr)$ time algorithm of [Björklund, Husfeldt, Koivisto, SICOMP 2009] that computes the chromatic number.

We resolve this long-standing open problem by generalizing and combining tools from the $(k+2)$-coloring to $k$-list-coloring reduction of [Zamir, ICALP 2021] and the hypergraph-containers based approach in [Zamir, STOC 2023]. Together with new algorithms for list-coloring instances mixing long and short color lists, this yields an iterable reduction from $(k+1)$-list-coloring to $k$-list-coloring over fixed palettes.

Subjects:

Data Structures and Algorithms (cs.DS)

Cite as:<br>arXiv:2607.25973 [cs.DS]

(or<br>arXiv:2607.25973v1 [cs.DS] for this version)

https://doi.org/10.48550/arXiv.2607.25973

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arXiv-issued DOI via DataCite (pending registration)

Submission history<br>From: Or Zamir [view email]<br>[v1]<br>Tue, 28 Jul 2026 16:53:58 UTC (47 KB)

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View a PDF of the paper titled k-Coloring is Faster than Computing the Chromatic Number, by Or Zamir<br>View PDF<br>HTML (experimental)<br>TeX Source

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