{"paper":{"title":"The Kelmans-Seymour conjecture III: 3-vertices in $K_4^-$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dawei He, Xingxing Yu, Yan Wang","submitted_at":"2016-09-19T14:40:15Z","abstract_excerpt":"Let $G$ be a 5-connected nonplanar graph and let $x_1,x_2,y_1,y_2\\in V(G)$ be distinct, such that $G[\\{x_1,x_2,y_1,y_2\\}]\\cong K_4^-$ and $y_1y_2\\notin E(G)$. We show that one of the following holds: $G-x_1$ contains $K_4^-$, or $G$ contains a $K_4^-$ in which $x_1$ is of degree 2, or $G$ contains a $TK_5$ in which $x_1$ is not a branch vertex, or $\\{x_2,y_1,y_2\\}$ may be chosen so that for any distinct $z_0, z_1\\in N(x_1)-\\{x_2,y_1,y_2\\}$, $G-\\{x_1v:v\\notin \\{z_0, z_1,x_2, y_1,y_2\\}\\}$ contains $TK_5$. This result will be used to prove the Kelmans-Seymour conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1609.05747","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"}