Geometrization of Graphs: Towards Bounding the Chromatic Number via High-Dimensional Embedding

TL;DR

This paper introduces a geometric framework transforming graphs into high-dimensional complexes to establish new bounds on their chromatic number based on embeddability criteria.

Contribution

It develops a systematic method to relate graph minors and embeddability of associated complexes to chromatic bounds, extending geometric and topological techniques.

Findings

Embeddability of the complex implies an upper bound on chromatic number.

Graphs excluding certain minors have complexes that embed into Euclidean space.

Extended Discharging method applied to high-dimensional coloring problems.

Abstract

We establish a geometric framework by transforming a graph $G$ into a $(d-1)$-dimensional CW complex $U^{d-1}(G)$. This construction is achieved by systematically attaching $i$-spheres ($2 \le i \le d-1$) to $G$ according to specific rules, ensuring that the $j$-th homotopy group of $U^{d-1}(G)$ are trivial for $j = 0, 1, \dots, d-2$. Building upon this construction, we provide a necessary and sufficient condition for $U^{d-1}(G)$ to be embeddable into $\mathbb{R}^d$, which yields an upper bound for the chromatic number $\chi(G)$. To be more specific, we prove that if $G$ does not contain $K_{d+3}$ and $K_{i, d+4-i}$ ($i \in \{2, 3, \dots, \lfloor \frac{d+4}{2} \rfloor \}$) as a minor, then $U^{d-1}(G)$ embeds into $\mathbb{R}^d$ and $\chi(G) \leq 3\cdot 2^{d-1}$. Finally, as a preliminary attempt, we extend the Discharging method to $\mathbb{R}^d$ and investigate the coloring problem for $(d-2)$-faces in $\mathbb{R}^d$.