{"paper":{"title":"Proof of a conjecture of Bauer, Fan and Veldman","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Tri Lai","submitted_at":"2013-09-20T20:58:37Z","abstract_excerpt":"For a 1-tough graph $G$ we define $\\sigma_3(G) = \\min\\{\\deg(u) + \\deg(v)+ \\deg(w):$ $\\{u, v, w\\}$ is an independent set of vertices$\\}$ and $NC2(G)=\\min \\{|N(u)\\cup N(v)|: d(u,v)=2\\}$. D. Bauer, G. Fan and H.J.Veldman proved that $c(G)\\geq \\min\\{n,2NC2(G)\\}$ for any 1-tough graph $G$ with $\\sigma_3(G)\\geq n\\geq 3$, where $c(G)$ is the circumference of $G$ (D. Bauer, G. Fan and H.J.Veldman,Hamiltonian properties of graphs with large neighborhood unions,Discrete Mathematics, 1991). They also conjectured a stronger upper bound for the circumference: $c(G)\\geq\\min\\{n,2NC2(G)+4\\}$.In this paper, we"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1309.5379","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"}