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.

翻译：在图$G$中，一个"支配性$K_t$-模型"是一个由$t$个两两顶点不交的连通子图构成的序列$(T_1,\dots,T_t)$，使得对于任意$1\leq i<j\leq t$，$T_j$中的每个顶点在$T_i$中均有一个邻点。将"$T_j$中的每个顶点"替换为"$T_j$中的某个顶点"即得到$K_t$-模型的标准定义，它与图$G$中的$K_t$- minors是等价的。我们证明：不含支配性$K_5$-模型的图是4-可着色的。这一结果推广并显著强于平面图或无$K_5$-子图图的4色定理，同时也推进了关于5-可着色图中$K_5$-细分问题的Hajós猜想。