Necessary Spectral Conditions for Coloring Hypergraphs

Hoffman proved that for a simple graph G, the chromatic number χ(G) obeys χ(G) ≥ 1 − λ1 λn where λ1 and λn are the maximal and minimal eigenvalues of the adjacency matrix of G respectively. Lovász later showed that χ(G) ≥ 1− λ1 λn 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 d̄ ≤ − 3 2 λmin where d̄ is the average degree and λmin 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.