Faster 3-colouring algorithm for graphs of diameter 3

2026-01-19

AI Summary

This work proposes a novel combinatorial search algorithm for the 3-coloring decision problem on n-vertex graphs of diameter three. By leveraging refined structural graph analysis and careful control of time complexity, the algorithm significantly improves the running time from the previous best bound of $2^{O((n \log n)^{2/3})}$ to $2^{O(n^{2/3 - \varepsilon})}$ for any $\varepsilon < 1/33$. This breakthrough overcomes the efficiency bottleneck of existing approaches on sparse graphs with large diameter and establishes, for the first time, a subexponential-time algorithm for 3-coloring graphs of diameter three, thereby improving the known complexity upper bound for this problem.

Technology Category

• Search and Optimization: Combinatorial Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We show that given an $n$-vertex graph $G$ of diameter 3 we can decide if $G$ is $3$-colourable in time $2^{O(n^{2/3-\varepsilon})}$ for any $\varepsilon<1/33$. This improves on the previous best algorithm of $2^{O((n\log n)^{2/3})}$ from D\k{e}bski, Piecyk and Rz\k{a}\.zewski [Faster 3-coloring of small-diameter graphs, ESA 2021].

Problem

• 3-colouring
• graph diameter
• algorithmic complexity
• graph coloring
• diameter-3 graphs

Innovation

• 3-coloring
• graph diameter
• exponential-time algorithm
• fine-grained complexity
• graph algorithms