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arXiv · 2605.10112

The Dominating 4-Colour Theorem

António Girão·Freddie Illingworth·Bojan Mohar·Sergey Norin·Raphael Steiner·Youri Tamitegama·Jane Tan·David R. Wood·Jung Hon Yip
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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 Haj\'{o}s' conjecture on $K_5$-subdivisions in $5$-chromatic graphs.

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António Girão, Freddie Illingworth, Bojan Mohar, Sergey Norin, Raphael Steiner, Youri Tamitegama, Jane Tan, David R. Wood, Jung Hon Yip. 2026-05-11. The Dominating 4-Colour Theorem. https://arxiv.org/abs/2605.10112

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