[2607.27159] Breaking the $2^n$ barrier for graph $k$-coloring
0$ such that graph $k$-coloring can be solved by a randomized algorithm with one-sided error in time $O((2-\varepsilon_k)^n)$. Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the $2^n \cdot \mathrm{poly}(n)$-time algorithm of Björklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for $k \le 6$."/>
0$ such that graph $k$-coloring can be solved by a randomized algorithm with one-sided error in time $O((2-\varepsilon_k)^n)$. Prior to this..."/>
0$ such that graph $k$-coloring can be solved by a randomized algorithm with one-sided error in time $O((2-\varepsilon_k)^n)$. Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the $2^n \cdot \mathrm{poly}(n)$-time algorithm of Bj\"orklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for $k \le 6$." />
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Computer Science > Data Structures and Algorithms
arXiv:2607.27159 (cs)
[Submitted on 29 Jul 2026]
Title:Breaking the $2^n$ barrier for graph $k$-coloring
Authors:Kevin Pratt<br>View a PDF of the paper titled Breaking the $2^n$ barrier for graph $k$-coloring, by Kevin Pratt
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Abstract:We show that for all $k$, there exists $\varepsilon_k > 0$ such that graph $k$-coloring can be solved by a randomized algorithm with one-sided error in time $O((2-\varepsilon_k)^n)$. Prior to this work and independent concurrent work of Zamir [arXiv, 2026], exponential improvements over the $2^n \cdot \mathrm{poly}(n)$-time algorithm of Björklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for $k \le 6$.
Subjects:
Data Structures and Algorithms (cs.DS)
Cite as:<br>arXiv:2607.27159 [cs.DS]
(or<br>arXiv:2607.27159v1 [cs.DS] for this version)
https://doi.org/10.48550/arXiv.2607.27159
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arXiv-issued DOI via DataCite (pending registration)
Submission history<br>From: Kevin Pratt [view email]<br>[v1]<br>Wed, 29 Jul 2026 17:36:11 UTC (16 KB)
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