{"paper":{"title":"Neighborhood complexes, homotopy test graphs and a contribution to a conjecture of Hedetniemi","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Samir Shukla","submitted_at":"2018-10-01T12:18:17Z","abstract_excerpt":"The neighborhood complex $\\N(G)$ of a graph $G$ were introduced by L. Lov{\\'a}sz in his proof of Kneser conjecture. He proved that for any graph $G$,\n  \\begin{align} \\label{abstract} \\chi(G) \\geq conn(\\N(G))+3.\n  \\end{align}\n  In this article we show that for a class of exponential graphs the bound given in (\\ref{abstract}) is sharp. Further, we show that the neighborhood complexes of these exponential graphs are spheres up to homotopy.\n  We were also able to find a class of exponential graphs, which are homotopy test graphs.\n  Hedetniemi's conjecture states that the chromatic number of the ca"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.00648","kind":"arxiv","version":2},"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"}