Breaking the $2^n$ barrier for graph $k$-coloring

Kevin Pratt

Breaking the $2^n$ barrier for graph $k$-coloring

Data Structures and Algorithms 2026-07-29 v1

Authors: Kevin Pratt

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 Bj\"orklund, Husfeldt, and Koivisto [SIAM Journal on Computing, 2009] were only known for $k \le 6$ .

Cite

@article{arxiv.2607.27159,
  title  = {Breaking the $2^n$ barrier for graph $k$-coloring},
  author = {Kevin Pratt},
  journal= {arXiv preprint arXiv:2607.27159},
  year   = {2026}
}