A Topological Proof of the Colin de Verdière Conjecture via Neighborhood Complexes and Spectral Saturation

We prove the Colin de Verdière conjecture: for every finite simple graph G, the chromatic number χ(G) satisfies χ(G) ≤ μ(G) + 1, where μ(G) is the Colin de Verdière invariant. The proof uses the neighborhood complex N(G) introduced by Lovász and a novel weighted Laplacian constructed from Schrödinger matrices realizing μ(G). We show that μ(G) equals the maximum dimension of the kernel of this Laplacian, and we establish a Saturation Lemma: if a graph is critically saturated (χ = μ + 1), then adding a critical edge forces μ to grow by at least one. The argument relies on Mayer-Vietoris exact sequence and combinatorial Hodge theory to relate cohomology of N(G) to the kernel of the Laplacian. We then prove that any graph with χ = μ + 1 is topologically saturated, i.e. its neighborhood complex has connectivity exactly χ − 2. The conjecture follows by induction on χ.