The Dominating 4-Colour Theorem

TL;DR

The paper proves that graphs without a dominating $K_5$-model are 4-colorable, extending the classic 4-color theorem and advancing understanding of $K_5$-subdivisions in chromatic graphs.

Contribution

It introduces the concept of dominating $K_t$-models and proves that graphs lacking a dominating $K_5$-model are 4-colorable, strengthening previous results.

Findings

Graphs with no dominating $K_5$-model are 4-colorable.

Generalizes the 4-color theorem beyond planar graphs.

Progresses towards Hajós' conjecture on $K_5$-subdivisions.

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.