{"paper":{"title":"Vizing's 2-factor Conjecture Involving Toughness and Maximum Degree Conditions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jinko Kanno, Songling Shan","submitted_at":"2017-09-05T19:21:00Z","abstract_excerpt":"Let $G$ be a simple graph, and let $\\Delta(G)$ and $\\chi'(G)$ denote the maximum degree and chromatic index of $G$, respectively. Vizing proved that $\\chi'(G)=\\Delta(G)$ or $\\Delta(G)+1$. We say $G$ is $\\Delta$-critical if $\\chi'(G)=\\Delta+1$ and $\\chi'(H)<\\chi'(G)$ for every proper subgraph $H$ of $G$. In 1968, Vizing conjectured that if $G$ is a $\\Delta$-critical graph, then $G$ has a 2-factor. Let $G$ be an $n$-vertex $\\Delta$-critical graph. It was proved that if $\\Delta(G)\\ge n/2$, then $G$ has a 2-factor; and that if $\\Delta(G)\\ge 2n/3+12$, then $G$ has a hamiltonian cycle, and thus a 2-"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.02241","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"}