{"paper":{"title":"Cycle-saturated graphs with minimum number of edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Younjin Kim, Zoltan Furedi","submitted_at":"2011-03-01T02:18:36Z","abstract_excerpt":"A graph $G$ is called $H$-saturated if it does not contain any copy of $H$, but for any edge $e$ in the complement of $G$ the graph $G+e$ contains some $H$. The minimum size of an $n$-vertex $H$-saturated graph is denoted by $\\sat(n,H)$. We prove $$\\sat(n,C_k) = n + n/k + O((n/k^2) + k^2)$$ holds for all $n\\geq k\\geq 3$, where $C_k$ is a cycle with length $k$. We have a similar result for semi-saturated graphs $$\\ssat(n,C_k) = n + n/(2k) + O((n/k^2) + k).$$ We conjecture that our three constructions are optimal."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.0067","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"}