The Dominating 4-Colour Theorem

2026-05-11

AI Summary

This study investigates whether graphs excluding a dominating $K_5$-model are 4-colorable, aiming to extend the Four Color Theorem and advance Hajós’ conjecture. The authors introduce the novel notion of a dominating $K_t$-model and combine techniques from subgraph decomposition, component connectivity analysis, and structural induction to prove that every graph without a dominating $K_5$-model is indeed 4-colorable. This result strictly strengthens both the classical Four Color Theorem—restricted to planar graphs—and known coloring theorems for graphs excluding a $K_5$-minor. Moreover, it provides significant progress toward Hajós’ conjecture, which posits that every non-4-colorable graph must contain a subdivision of $K_5$.

Abstract

A "dominating $K_t$-model" in a graph $G$ is a sequence $(T_1,\dots,T_t)$ of pairwise vertex-disjoint connected subgraphs of $G$, such that whenever $1\leq i<j\leq t$ every vertex in $T_j$ has a neighbour in $T_i$. Replacing "every vertex in $T_j$" by "some vertex in $T_j$" retrieves the standard definition of $K_t$-model, which is equivalent to a $K_t$-minor in $G$. We prove that every graph with no dominating $K_5$-model is $4$-colourable. This generalises and is significantly stronger than the 4-colour theorem for planar graphs or for graphs with no $K_5$-minor. It also makes progress towards Hajós' conjecture on $K_5$-subdivisions in $5$-chromatic graphs.

Problem

• dominating K_t-model
• graph coloring
• 4-colourable
• K_5-minor
• Hajós' conjecture

Innovation

• dominating K_t-model
• 4-colourability
• graph minor
• Hajós' conjecture
• graph coloring