{"paper":{"title":"On saturation problems involving clique number and matching number","license":"http://creativecommons.org/licenses/by-nc-sa/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Guorong Gao, Jianfeng Hou, Yue Ma, Zian Chen","submitted_at":"2026-06-08T16:59:24Z","abstract_excerpt":"For a clique $K_r$, a graph is $K_r$-saturated if it contains no copy of $K_r$ and the addition of any edge from its complement creates a $K_r$. A classical result of Erd\\H{o}s-Hajnal-Moon and Zykov shows that the number of edges of an $n$-vertex $K_r$-saturated graph is at least $(r-2)n-\\binom{r-1}{2}$. In this paper, we focus on the number of edges of the $K_r$-saturated graphs with a fixed matching number. Let $G$ be an $n$-vertex $K_r$-saturated graph with matching number $\\nu(G) = s$. For sufficiently large $n$, we prove that the number of edges \\begin{equation*}\n  e(G)\\geq \\left\\{\\begin{"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.09733","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.09733/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}