{"paper":{"title":"Hadwiger's conjecture for graphs with forbidden holes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Brian Thomas, Zi-Xia Song","submitted_at":"2016-07-22T15:57:40Z","abstract_excerpt":"Given a graph $G$, the Hadwiger number of $G$, denoted by $h(G)$, is the largest integer $k$ such that $G$ contains the complete graph $K_k$ as a minor. A hole in $G$ is an induced cycle of length at least four. Hadwiger's Conjecture from 1943 states that for every graph $G$, $h(G)\\ge \\chi(G)$, where $\\chi(G)$ denotes the chromatic number of $G$. In this paper we establish more evidence for Hadwiger's conjecture by showing that if a graph $G$ with independence number $\\alpha(G)\\ge3$ has no hole of length between $4$ and $2\\alpha(G)-1$, then $h(G)\\ge\\chi(G)$. We also prove that if a graph $G$ w"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.06718","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"}