The Dominating 4-Colour Theorem
2026-05-11 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors study a special way of finding certain structures called dominating K_t-models in graphs, which are groups of connected parts with strong neighbor connections. They show that if a graph does not have a dominating K_5-model, then it can be colored using only four colors without any two adjacent vertices sharing the same color. This result extends the famous 4-color theorem beyond planar graphs and graphs without K_5-minors. Their work also contributes to progress on a conjecture related to coloring graphs that contain specific K_5 subdivisions.
graph theorydominating K_t-modelgraph coloring4-color theoremplanar graphK_5-minorminorsubdivisionHajós' conjecturechromatic number
Authors
António Girão, Freddie Illingworth, Bojan Mohar, Sergey Norin, Raphael Steiner, Youri Tamitegama, Jane Tan, David R. Wood, Jung Hon Yip
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.