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.