The Dominating 4-Colour Theorem
This result strengthens the 4-color theorem to a broader class of graphs, advancing graph coloring theory for structural graph theorists.
The authors prove that every graph without a dominating K5-model is 4-colorable, generalizing the 4-color theorem for planar graphs and graphs with no K5-minor, and making progress toward Hajós' conjecture.
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.