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Hoffman proved that for a simple graph \( G \), the chromatic number \( \chi(G) \) obeys \( \chi(G) \geq 1 – \frac{\lambda_1}{\lambda_n} \), where \( \lambda_1 \) and \( \lambda_n \) are the maximal and minimal eigenvalues of the adjacency matrix of \( G \), respectively. Lovász later showed that \( \chi(G) \geq 1 – \frac{\lambda_1}{\lambda_n} \) for any (perhaps negatively) weighted adjacency matrix.
In this paper, we give a probabilistic proof of Lovász’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 \( \bar{d} \leq – \frac{3}{2} \lambda_{\text{min}} \) where \( \bar{d} \) is the average degree and \( \lambda_{\text{min}} \) 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.