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It is natural to study how far a $ C_{2k+1}$-free graph is from being bipartite.Let $T^*(r, n)$ be obtained by adding a suspension $K_{r}$ with $1$ suspension point to $K_{\\lfloor\\frac{n-r+1}{2}\\rfloor, \\lceil\\frac{n-r+1}{2}\\rceil}$. We show that for integers $r, k$ with $3\\le r\\le 2k-4$ and $n\\ge 20(r+2)^2k$, if $G$ is a $C_{2k+1}$-free $n$-vertex graph with $e(G)\\ge e(T^*(r, n))$, then $G$ is obtained by adding suspensions to a bipartite graph one "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2408.15487","kind":"arxiv","version":2},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2024-08-28T02:21:37Z","cross_cats_sorted":[],"title_canon_sha256":"3c148a67a34f1c8e32cdff8005d845d7d9c9b6da5e5d73e70a58eef8224b5cc0","abstract_canon_sha256":"274927062c5c5051a5eed3bd76b03f7016c98165875ba416425d3484840e0156"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:23:57.494890Z","signature_b64":"aAH29oZacP4RiN1TY1sPOQWg8eP3t40/zUoOohxIrxXiQVF0rIU407N/Z0nZ/GGBpCweEpNmmW9m5WZzYiLSCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4488f0d90a0746e05aa334afed3f3693481344389ba999abadb67e96f614f95c","last_reissued_at":"2026-07-05T09:23:57.492366Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:23:57.492366Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A strong structural stability of $C_{2k+1}$-free graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Yuejian Peng, Zilong Yan","submitted_at":"2024-08-28T02:21:37Z","abstract_excerpt":"F\\\"uredi and Gunderson showed that $ex(n, C_{2k+1})$ is achieved only on $K_{\\lfloor\\frac{n}{2}\\rfloor, \\lceil\\frac{n}{2}\\rceil}$ if $n\\ge 4k-2$. It is natural to study how far a $ C_{2k+1}$-free graph is from being bipartite.Let $T^*(r, n)$ be obtained by adding a suspension $K_{r}$ with $1$ suspension point to $K_{\\lfloor\\frac{n-r+1}{2}\\rfloor, \\lceil\\frac{n-r+1}{2}\\rceil}$. We show that for integers $r, k$ with $3\\le r\\le 2k-4$ and $n\\ge 20(r+2)^2k$, if $G$ is a $C_{2k+1}$-free $n$-vertex graph with $e(G)\\ge e(T^*(r, n))$, then $G$ is obtained by adding suspensions to a bipartite graph one "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2408.15487","kind":"arxiv","version":2},"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/2408.15487/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2408.15487","created_at":"2026-07-05T09:23:57.493862+00:00"},{"alias_kind":"arxiv_version","alias_value":"2408.15487v2","created_at":"2026-07-05T09:23:57.493862+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2408.15487","created_at":"2026-07-05T09:23:57.493862+00:00"},{"alias_kind":"pith_short_12","alias_value":"ISEPBWIKA5DO","created_at":"2026-07-05T09:23:57.493862+00:00"},{"alias_kind":"pith_short_16","alias_value":"ISEPBWIKA5DOAWVD","created_at":"2026-07-05T09:23:57.493862+00:00"},{"alias_kind":"pith_short_8","alias_value":"ISEPBWIK","created_at":"2026-07-05T09:23:57.493862+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":2,"internal_anchor_count":0,"sample":[{"citing_arxiv_id":"2604.09394","citing_title":"On the chromatic profile for tripartite graphs and beyond","ref_index":28,"is_internal_anchor":false},{"citing_arxiv_id":"2604.09394","citing_title":"On the chromatic profile for tripartite graphs and beyond","ref_index":28,"is_internal_anchor":false}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN","json":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN.json","graph_json":"https://pith.science/api/pith-number/ISEPBWIKA5DOAWVDGSX62PZWSN/graph.json","events_json":"https://pith.science/api/pith-number/ISEPBWIKA5DOAWVDGSX62PZWSN/events.json","paper":"https://pith.science/paper/ISEPBWIK"},"agent_actions":{"view_html":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN","download_json":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN.json","view_paper":"https://pith.science/paper/ISEPBWIK","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2408.15487&json=true","fetch_graph":"https://pith.science/api/pith-number/ISEPBWIKA5DOAWVDGSX62PZWSN/graph.json","fetch_events":"https://pith.science/api/pith-number/ISEPBWIKA5DOAWVDGSX62PZWSN/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN/action/storage_attestation","attest_author":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN/action/author_attestation","sign_citation":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN/action/citation_signature","submit_replication":"https://pith.science/pith/ISEPBWIKA5DOAWVDGSX62PZWSN/action/replication_record"}},"created_at":"2026-07-05T09:23:57.493862+00:00","updated_at":"2026-07-05T09:23:57.493862+00:00"}