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

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 $χ(G)$. 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 $χ(G) \leq 3\cdot 2^{d-1}$. Finally, as a preliminary attempt, we extend the Discharging method to $\mathbb{R}^d$ and investigate the coloring problem for $(d-2)$-faces in $\mathbb{R}^d$.

💡 Research Summary

The paper introduces a novel geometric framework that lifts a finite, simple, connected graph G into a (d‑1)‑dimensional CW complex denoted U^{d‑1}(G). The construction proceeds by systematically attaching i‑spheres (for 2 ≤ i ≤ d‑1) to G, each i‑sphere being realized as the boundary of an (i+1)‑ball. Two notions of cycles are distinguished: induced cycles (or induced i‑spheres) whose removal leaves the underlying complex connected, and chordless cycles (or chordless i‑spheres) which contain no higher‑dimensional chords. By filling only induced cycles with disks the authors preserve the 1‑skeleton of G while raising its dimension, thereby creating a (d‑1)‑dimensional topological hypergraph.

A key topological property of the constructed complex is that it is d‑2‑connected: all homotopy groups π_j(U^{d‑1}(G)) vanish for j = 0,1,…,d‑2. This high level of connectivity is essential for embedding results in higher dimensions. The authors then define a class of complexes called R^{d}‑hypergraphs. An R^{d}‑hypergraph is a (d‑1)‑dimensional topological hypergraph in which each (d‑1)-dimensional hyperedge is homeomorphic to a (d‑1)-simplex, and any two hyperedges intersect only along common i‑faces with i ≤ d‑2. This mirrors the planar graph characterization (Kuratowski/Wagner) but in ℝ^{d}.

The central theorem (Theorem 1.1) provides a necessary and sufficient condition for U^{d‑1}(G) to embed in ℝ^{d}. For d = 3 the forbidden minors are K₆ and K_{3,4}; for d = 4 they are K₇, K_{4,4}, and K₃ ∧ (K₁ ∪ K₄); for d ≥ 5 a finite list of minors (given in Table 1) must be avoided. Equivalently, if G does not contain K_{d+3} nor any bipartite complete graph K_{i,d+4‑i} with 2 ≤ i ≤ ⌊(d+4)/2⌋ as a minor, then U^{d‑1}(G) embeds in ℝ^{d}.

Having established embeddability, the authors turn to coloring. In an R^{d}‑hypergraph the problem of coloring the (d‑2)-faces is equivalent to vertex‑coloring the original graph G. By extending the discharging method to ℝ^{d}, they assign “charges” to (d‑2)-faces and redistribute them according to local structural rules (bridges, ear decompositions, S‑components). This analysis yields the explicit chromatic bound

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