k-Coloring is Faster than Computing the Chromatic Number

2026-07-28

AI Summary

This work addresses the graph k-coloring problem for fixed k, aiming to break the long-standing O*(2ⁿ) time complexity barrier. By integrating Zamir’s reduction from (k+2)-coloring to k-list coloring, the hypergraph container method, and a novel list-coloring algorithm tailored for handling mixed instances with both short and long color lists, the authors present the first randomized algorithm running in (2−εₖ)ⁿ time for every fixed k, where εₖ > 0. This result extends exponential-speedup algorithms—previously known only for k ≤ 6—to all fixed values of k, thereby resolving a major open problem in the field.

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.

Problem

• k-coloring
• chromatic number
• exponential-time algorithms
• graph coloring
• randomized algorithm

Innovation

• k-coloring
• list-coloring
• hypergraph containers
• iterable reduction
• randomized algorithm