{"paper":{"title":"Necessary Spectral Conditions for Coloring Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Franklin H. J. Kenter","submitted_at":"2014-12-11T23:00:37Z","abstract_excerpt":"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.\n  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 sho"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1412.3855","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}