{"paper":{"title":"Connectedness of Certain Graph Coloring Complexes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Nandini Nilakantan, Samir Shukla","submitted_at":"2017-02-12T13:36:10Z","abstract_excerpt":"In this article, we consider the bipartite graphs $K_2 \\times K_n$. We prove that the connectedness of the complex $\\displaystyle \\text{Hom}(K_2\\times K_{n}, K_m) $ is $m-n-1$ if $m \\geq n$ and $m-3$ in the other cases. Therefore, we show that for this class of graphs, $\\text{Hom} (G, K_m)$ is exactly $m-d-2$ connected, $m \\geq n$, where $d$ is the maximal degree of the graph $G$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.03527","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"}