{"paper":{"title":"Two problems on booksize and triangular edges in Nosal graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jing Zeng, Lihua You, Xiaoxue Zhang, Xinghui Zhao","submitted_at":"2026-07-16T14:43:37Z","abstract_excerpt":"A graph $G$ with $m$ edges is said to be a Nosal graph if $\\rho(G)>\\sqrt{m}$. For a graph $G$, we write $bk(G)$ for its maximum book size and $\\tau(G)$ for the number of edges contained in triangles. Li, Liu and Zhang [J. Combin. Theory Ser. B 179 (2026) 219--249] proved that every $m$-edge Nosal graph satisfies $bk(G)> \\frac{1}{24}\\sqrt{m}$ and $\\tau(G) > \\frac{1}{12}\\sqrt{m}$. Recently, Zhai, Li and Lou [arXiv:2601.10163v2] proved that every $m$-edge Nosal graph satisfies $bk(G)> \\frac{1}{9}\\sqrt{m}$. In this paper, we establish the following result: Every $m$-edge graph $G$ with no isolated"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.15071","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/2607.15071/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"}