{"paper":{"title":"The edge spectrum of $K_4^-$-saturated graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jun Gao, Xinmin Hou, Yue Ma","submitted_at":"2018-04-27T06:49:23Z","abstract_excerpt":"Given graphs $G$ and $H$, $G$ is $H$-saturated if $G$ does not contain a copy of $H$ but the addition of any edge $e\\notin E(G)$ creates at least one copy of $H$ within $G$. The edge spectrum of $H$ is the set of all possible sizes of an $H$-saturated graph on $n$ vertices. Let $K_4^-$ be a graph obtained from $K_4$ by deleting an edge. In this note, we show that (a) if $G$ is a $K_4^-$-saturated graph with $|V(G)|=n$ and $|E(G)|>\\lfloor \\frac{n-1}{2} \\rfloor \\lceil \\frac{n-1}{2} \\rceil +2$, then $G$ must be a bipartite graph; (b) there exists a $K_4^-$-saturated non-bipartite graph on $n\\ge 1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1804.10359","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"}