{"paper":{"title":"A sharp Dirac-Erd\\H{o}s type bound for large graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr V. Kostochka, Andrew McConvey, Henry A. Kierstead","submitted_at":"2017-07-12T20:07:08Z","abstract_excerpt":"Let $k \\geq 3$ be an integer, $h_{k}(G)$ be the number of vertices of degree at least $2k$ in a graph $G$, and $\\ell_{k}(G)$ be the number of vertices of degree at most $2k-2$ in $G$. Dirac and Erd\\H{o}s proved in 1963 that if $h_{k}(G) - \\ell_{k}(G) \\geq k^{2} + 2k - 4$, then $G$ contains $k$ vertex-disjoint cycles. For each $k\\geq 2$, they also showed an infinite sequence of graphs $G_k(n)$ with $h_{k}(G_k(n)) - \\ell_{k}(G_k(n)) = 2k-1$ such that $G_k(n)$ does not have $k$ disjoint cycles. Recently, the authors proved that, for $k \\geq 2$, a bound of $3k$ is sufficient to guarantee the exist"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.03892","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"}