Bounding the Chromatic Number via High Dimensional Embedding

TL;DR

The paper proposes a high-dimensional program to relate the Hadwiger conjecture to graph embeddings in $\mathbb{R}^d$ by geometrizing graphs through the dimension-raising map $U^{d-1}(G)$, producing a $(d-1)$-dimensional $\mathbb{R}^d$-hypergraph. It proves a key sufficiency: if $G$ contains no $K_{d+3}$ or $K_{3,d+1}$ minors, then $U^{d-1}(G)$ embeds in $\mathbb{R}^d$ and $\chi(G) \le d(d+1)$, and it extends the Discharging method to higher dimensions to bound colorability of $d$-uniform $\mathbb{R}^d$-hypergraphs. The work develops a rich toolkit—bridges, hyper ear decompositions, $S$-components, and triangulations—that generalizes planar graph theory to $\mathbb{R}^d$, enabling a new pathway to generalize planar coloring results like the Four Color Theorem to higher dimensions. Collectively, these results link minor-closed structure, embeddability, and coloring in a unified higher-dimensional framework with potential implications for Hadwiger-type conjectures.

Abstract

A geometrization method that transforms a $(d+1)$-connected graph $G$ into a $(d-1)$-dimensional manifold $U^{d-1}(G)$ is first established through adding some $i$-balls with $2\le i \le d-1$ into $G$ such that the $j$-th homotopy group is trivial for $j=0, 1, \dots, d-2$. On this basis, we establish a sufficient condition for $U^{d-1}(G)$ to be embedded into $\mathbb{R}^d$ and an upper bound for $χ(G)$, the chromatic number of $G$. To be more specific, we prove that if $G$ contains neither $K_{d+3}$ nor $K_{3,d+1}$ as a minor, then $U^{d-1}(G)$ embeds into $\mathbb{R}^d$ and $χ(G) \leq d(d+1)$. Furthermore, based on the above theorem, we extend the Discharging method, originally developed for the study of the four color theorem, to $\mathbb{R}^d$. This generalized approach can be applied to investigate the coloring problems in $\mathbb{R}^d$.

Figures (14)

• Figure 1: Ear decomposition.
• Figure 2: A $2$-dimensional topological hypergraph.
• Figure 3: $B_1$ and $B_2$ are skew of $S^2$.
• Figure 4: An example of $S$-decomposition and marked $S$-decomposition of $\mathbb{R}^3$-hypergraph.
• Figure 5: Infinite graph $G$.

Theorems & Definitions (75)

• Definition 1: induced $i$-sphere
• Theorem 1.1
• Theorem 1.2
• Definition 2: $i$-face of an $n$-dimensional polytope hatcher2002algebraic
• Definition 3: $(d-1)$-dimensional topological hypergraph
• Definition 4: $\mathbb{R}^d$-hypergraph and non-$\mathbb{R}^d$-hypergraph
• Lemma 2.1
• proof
• Lemma 2.2
• Definition 5: multiple topological hyperedges