The Dominating 4-Colour Theorem
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arXiv
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| Main Authors: | , , , , , , , , |
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| Format: | Preprint |
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2026
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| author | Girão, António Illingworth, Freddie Mohar, Bojan Norin, Sergey Steiner, Raphael Tamitegama, Youri Tan, Jane Wood, David R. Yip, Jung Hon |
| author_facet | Girão, António Illingworth, Freddie Mohar, Bojan Norin, Sergey Steiner, Raphael Tamitegama, Youri Tan, Jane Wood, David R. Yip, Jung Hon |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_10112 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Dominating 4-Colour Theorem Girão, António Illingworth, Freddie Mohar, Bojan Norin, Sergey Steiner, Raphael Tamitegama, Youri Tan, Jane Wood, David R. Yip, Jung Hon Combinatorics Discrete Mathematics 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. |
| title | The Dominating 4-Colour Theorem |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2605.10112 |