Necessary Spectral Conditions for Coloring Hypergraphs

Hoffman proved that for a simple graph $G$, the chromatic number $\chi(G)$ obeys $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ where $\lambda_1$ and $\lambda_n$ are the maximal and minimal eigenvalues of the adjacency matrix of $G$ respectively. Lov'asz later showed that $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ for any (perhaps negatively) weighted adjacency matrix. In this paper, we give a probabilistic proof of Lov'asz’s theorem, then extend the technique to derive generalizations of Hoffman’s theorem when allowed a certain proportion of edge-conflicts. Using this result, we show that if a 3-uniform hypergraph is 2-colorable, then $\bar d \le -\frac{3}{2}\lambda_{\min}$ where $\bar d$ is the average degree and $\lambda_{\min}$ is the minimal eigenvalue of the underlying graph. We generalize this further for $k$-uniform hypergraphs, for the cases $k=4$ and $5$, by considering several variants of the underlying graph.

💡 Research Summary

The paper revisits a classic result in spectral graph theory – the Hoffman–Lovász bound on the chromatic number – and provides a fresh probabilistic proof that works for any weighted adjacency matrix, even with negative entries. Starting from the well‑known inequality
 χ(G) ≥ 1 − λ_max/λ_min,
the author constructs a random vector by assigning each vertex a complex k‑th root of unity according to a proper k‑coloring, and then multiplies this by a unit eigenvector corresponding to the largest eigenvalue λ_max. By applying the Courant–Fischer variational principle to the expectation of the Rayleigh quotient, the bound emerges naturally.

The novelty lies in extending this argument to a setting where a small proportion p of edges are allowed to be monochromatic (i.e., the coloring may contain conflicts). The same probabilistic machinery yields the inequality
 p ≥ ( \bar d + k λ_min − λ_min ) / (k \bar d),
where \bar d is the average degree of the graph. When p = 0 the inequality collapses to Hoffman’s original bound; when p = 1 it trivially permits a single‑coloring. This “conflict‑tolerant” version provides a bridge between exact colorability and approximate colorings.

To apply these ideas to hypergraphs, the author adopts the “second approach” to hypergraph spectra: rather than working with high‑order tensors, one studies several derived ordinary graphs. The first derived graph, called the underlying graph G(H), connects two vertices whenever they appear together in some hyperedge (multiple edges are allowed, reflecting multiplicities). The second family consists of s‑subset graphs G^{(s)}(H), whose vertices are s‑element subsets of the original vertex set; two such subsets are adjacent if they are disjoint and together lie inside a common hyperedge.

For a 3‑uniform hypergraph H that is 2‑colorable, each hyperedge must contain two vertices of one color and one of the other. Consequently, in G(H) at most one‑third of the edges can be monochromatic. Plugging this p = 1/3 into the conflict‑tolerant inequality (with k = 2) yields the clean spectral condition
 \bar d ≤ −(3/2) λ_min,
where λ_min is the smallest eigenvalue of the adjacency matrix of G(H). This is Theorem 2.

The analysis for 4‑ and 5‑uniform hypergraphs is more intricate because hyperedges can be split as (2,2), (3,1) or (4,0) (and analogous splits for 5‑uniform). The author introduces a parameter p that measures the fraction of hyperedges that are “unbalanced” (e.g., three vertices of one color and one of the other). By counting how many underlying edges become monochromatic under a given 2‑coloring, the author derives a bound on p in terms of the graph spectra. Simultaneously, the 2‑subset graph G^{(2)}(H) is examined; its average degree is a simple function of \bar d and n, and its smallest eigenvalue λ^{(2)}_min can be bounded using known results for strongly regular graphs. Combining the two inequalities eliminates p and yields the final bounds:

• For 4‑uniform hypergraphs (Theorem 3):
   \bar d ≤ −2 λ_min −