{"paper":{"title":"Coloring $\\Delta$-Critical Graphs With Small High Vertex Cliques","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Landon Rabern","submitted_at":"2011-02-04T21:48:30Z","abstract_excerpt":"We prove that $K_{\\chi(G)}$ is the only critical graph $G$ with $\\chi(G) \\geq \\Delta(G) \\geq 6$ and $\\omega(\\mathcal{H}(G)) \\leq \\left \\lfloor \\frac{\\Delta(G)}{2} \\right \\rfloor - 2$. Here $\\mathcal{H}(G)$ is the subgraph of $G$ induced on the vertices of degree at least $\\chi(G)$. Setting $\\omega(\\mathcal{H}(G)) = 1$ proves a conjecture of Kierstead and Kostochka."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1102.1023","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"}