3-Colouring Graphs Excluding a Fixed Minor

2026-07-02

AI Summary

This work addresses the clustered 3-coloring problem for graph classes excluding a fixed graph $H$ as a minor, aiming to minimize the size of monochromatic connected components. By integrating graph minor theory, structural graph theory, and combinatorial optimization techniques, the study extends prior results on planar graphs to all proper minor-closed graph classes. It constructively proves that any $n$-vertex graph excluding $H$ as a minor admits a 3-coloring in which every monochromatic connected component has size $O_H(n^{4/9})$. This bound significantly improves upon the previously known general upper bound of $O(\sqrt{n})$, demonstrating both the breakthrough nature and broad applicability of the proposed approach.

Abstract

We show that, for every fixed graph $H$, every $n$-vertex graph $G$ that excludes $H$ as a minor is $3$-colourable with clustering $O_H(n^{4/9})$. That is, there exists a function $f$ such that for every graph $H$, every $n\ge 1$, every $n$-vertex graph $G$ that excludes $H$ as a minor has a vertex colouring with $3$ colours in which each monochromatic component has size at most $f(H)\cdot n^{4/9}$. This generalizes a recent result of Dujmović, Morin, Norin, and Wood (\textit{arXiv}:2507.03163) from planar graphs to all proper minor-closed graph classes and is the first improvement on clustered $3$-colouring of proper minor-closed graph classes since the upper bound of $O_H(\sqrt{n})$ due to Linial, Matoušek, Sheffet, and Tardos (\textit{Comb. Prob. Comput.}, \textbf{17}(4):577--589, 2008).

Problem

• 3-colouring
• graph minor
• clustering
• minor-closed graph classes
• monochromatic component

Innovation

• clustered coloring
• graph minor
• 3-colouring
• minor-closed graph classes
• monochromatic component