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For an integer $k\\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given graph $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $m\\ge4$ vertices and let $\\Theta_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. We prove that $GR_k(\\Theta_{2n+1"},"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":"1809.00227","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-01T17:42:35Z","cross_cats_sorted":[],"title_canon_sha256":"6cbcdc73badb77e6490da9e331b90b6015e7411254097a6f6222f0ded2ec3115","abstract_canon_sha256":"53cb8699c316a63dfd8ab67e7588bf9f7aa90890d792df9e9e6fa4bf1e5645e5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T01:35:42.812053Z","signature_b64":"a+eemKULokJhv9a2hiw7ate2iuzaRQQ8MXnyEik7W1m74Qfwwce68su2Rp0evEBVzaAxz9gwGcED9QtigIdsDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9c1183fc12b2e7363f0f63618a99580bc76c48b5ebbd7bb7a0a7d040bd8accd1","last_reissued_at":"2026-07-05T01:35:42.811671Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T01:35:42.811671Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Gallai-Ramsey number of odd cycles with chords","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Fangfang Zhang, Yaojun Chen, Zi-Xia Song","submitted_at":"2018-09-01T17:42:35Z","abstract_excerpt":"A Gallai coloring is a coloring of the edges of a complete graph without rainbow triangles, and a Gallai $k$-coloring is a Gallai coloring that uses at most $k$ colors. For an integer $k\\geq 1$, the Gallai-Ramsey number $GR_k(H)$ of a given graph $H$ is the least positive integer $N$ such that every Gallai $k$-coloring of the complete graph $K_N$ contains a monochromatic copy of $H$. Let $C_m$ denote the cycle on $m\\ge4$ vertices and let $\\Theta_m$ denote the family of graphs obtained from $C_m$ by adding an additional edge joining two non-consecutive vertices. We prove that $GR_k(\\Theta_{2n+1"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00227","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/1809.00227/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":"1809.00227","created_at":"2026-07-05T01:35:42.811733+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.00227v2","created_at":"2026-07-05T01:35:42.811733+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00227","created_at":"2026-07-05T01:35:42.811733+00:00"},{"alias_kind":"pith_short_12","alias_value":"TQIYH7ASWLTT","created_at":"2026-07-05T01:35:42.811733+00:00"},{"alias_kind":"pith_short_16","alias_value":"TQIYH7ASWLTTMPYP","created_at":"2026-07-05T01:35:42.811733+00:00"},{"alias_kind":"pith_short_8","alias_value":"TQIYH7AS","created_at":"2026-07-05T01:35:42.811733+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP","json":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP.json","graph_json":"https://pith.science/api/pith-number/TQIYH7ASWLTTMPYPMNQYVGKYBP/graph.json","events_json":"https://pith.science/api/pith-number/TQIYH7ASWLTTMPYPMNQYVGKYBP/events.json","paper":"https://pith.science/paper/TQIYH7AS"},"agent_actions":{"view_html":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP","download_json":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP.json","view_paper":"https://pith.science/paper/TQIYH7AS","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.00227&json=true","fetch_graph":"https://pith.science/api/pith-number/TQIYH7ASWLTTMPYPMNQYVGKYBP/graph.json","fetch_events":"https://pith.science/api/pith-number/TQIYH7ASWLTTMPYPMNQYVGKYBP/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP/action/timestamp_anchor","attest_storage":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP/action/storage_attestation","attest_author":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP/action/author_attestation","sign_citation":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP/action/citation_signature","submit_replication":"https://pith.science/pith/TQIYH7ASWLTTMPYPMNQYVGKYBP/action/replication_record"}},"created_at":"2026-07-05T01:35:42.811733+00:00","updated_at":"2026-07-05T01:35:42.811733+00:00"}