k-Coloring is Faster than Computing the Chromatic Number

Or Zamir

k-Coloring is Faster than Computing the Chromatic Number

Data Structures and Algorithms 2026-07-28 v1

Authors: Or Zamir

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 [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.

Cite

@article{arxiv.2607.25973,
  title  = {k-Coloring is Faster than Computing the Chromatic Number},
  author = {Or Zamir},
  journal= {arXiv preprint arXiv:2607.25973},
  year   = {2026}
}