{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:SN7ZOWA2EAAI46NG2ARFPZWTFH","short_pith_number":"pith:SN7ZOWA2","schema_version":"1.0","canonical_sha256":"937f97581a20008e79a6d02257e6d329cdef7399fcd8b94ae7759116579145cb","source":{"kind":"arxiv","id":"1809.00739","version":1},"attestation_state":"computed","paper":{"title":"An Optimal $\\chi$-Bound for ($P_6$, diamond)-Free Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Kathie Cameron, Owen Merkel, Shenwei Huang","submitted_at":"2018-09-03T23:01:01Z","abstract_excerpt":"Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices and $K_t$ be the complete graph on $t$ vertices. The diamond is the graph obtained from $K_4$ by removing an edge. In this paper we show that every ($P_6$, diamond)-free graph $G$ satisfies $\\chi(G)\\le \\omega(G)+3$, where $\\chi(G)$ and $\\omega(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the famous 27-vertex Schl\\\"afli graph. Our result unifies previously known "},"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":"1809.00739","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-09-03T23:01:01Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"09a376920125204298544947a099bf766f67eed5a6114a5dd00b967fd14cba55","abstract_canon_sha256":"d64dffe2033e7d9d46c4e3ceccec3153a256e6e591d2dbe8be9f25ea50c2d3af"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:32.610397Z","signature_b64":"VYDM7385fGxNr9hM592g5TkKr/4zzNwpyD6BBhIUaljhohcBgtiO+mefXzBfmpNZRul1blnJZcH6eDN0wpBtDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"937f97581a20008e79a6d02257e6d329cdef7399fcd8b94ae7759116579145cb","last_reissued_at":"2026-05-18T00:06:32.609907Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:32.609907Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An Optimal $\\chi$-Bound for ($P_6$, diamond)-Free Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Kathie Cameron, Owen Merkel, Shenwei Huang","submitted_at":"2018-09-03T23:01:01Z","abstract_excerpt":"Given two graphs $H_1$ and $H_2$, a graph $G$ is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ or $H_2$. Let $P_t$ be the path on $t$ vertices and $K_t$ be the complete graph on $t$ vertices. The diamond is the graph obtained from $K_4$ by removing an edge. In this paper we show that every ($P_6$, diamond)-free graph $G$ satisfies $\\chi(G)\\le \\omega(G)+3$, where $\\chi(G)$ and $\\omega(G)$ are the chromatic number and clique number of $G$, respectively. Our bound is attained by the complement of the famous 27-vertex Schl\\\"afli graph. Our result unifies previously known "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1809.00739","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1809.00739","created_at":"2026-05-18T00:06:32.609987+00:00"},{"alias_kind":"arxiv_version","alias_value":"1809.00739v1","created_at":"2026-05-18T00:06:32.609987+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1809.00739","created_at":"2026-05-18T00:06:32.609987+00:00"},{"alias_kind":"pith_short_12","alias_value":"SN7ZOWA2EAAI","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_16","alias_value":"SN7ZOWA2EAAI46NG","created_at":"2026-05-18T12:32:53.628368+00:00"},{"alias_kind":"pith_short_8","alias_value":"SN7ZOWA2","created_at":"2026-05-18T12:32:53.628368+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"1907.05018","citing_title":"Structural domination and coloring of some ($P_7, C_7$)-free graphs","ref_index":7,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH","json":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH.json","graph_json":"https://pith.science/api/pith-number/SN7ZOWA2EAAI46NG2ARFPZWTFH/graph.json","events_json":"https://pith.science/api/pith-number/SN7ZOWA2EAAI46NG2ARFPZWTFH/events.json","paper":"https://pith.science/paper/SN7ZOWA2"},"agent_actions":{"view_html":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH","download_json":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH.json","view_paper":"https://pith.science/paper/SN7ZOWA2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1809.00739&json=true","fetch_graph":"https://pith.science/api/pith-number/SN7ZOWA2EAAI46NG2ARFPZWTFH/graph.json","fetch_events":"https://pith.science/api/pith-number/SN7ZOWA2EAAI46NG2ARFPZWTFH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH/action/storage_attestation","attest_author":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH/action/author_attestation","sign_citation":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH/action/citation_signature","submit_replication":"https://pith.science/pith/SN7ZOWA2EAAI46NG2ARFPZWTFH/action/replication_record"}},"created_at":"2026-05-18T00:06:32.609987+00:00","updated_at":"2026-05-18T00:06:32.609987+00:00"}