{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2023:BG7HTMTQ2XC2KRWHPWO5Q5KIIW","short_pith_number":"pith:BG7HTMTQ","schema_version":"1.0","canonical_sha256":"09be79b270d5c5a546c77d9dd87548459dc947b225f011f96e42150b289b14fe","source":{"kind":"arxiv","id":"2305.11757","version":1},"attestation_state":"computed","paper":{"title":"Linear $\\chi$-binding functions for $\\{P_3\\cup P_2, gem\\}$-free graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Athmakoori Prashant, M. Gokulnath, S. Francis Raj","submitted_at":"2023-05-19T15:44:14Z","abstract_excerpt":"Finding families that admit a linear $\\chi$-binding function is a problem that has interested researchers for a long time. Recently, the question of finding linear subfamilies of $2K_2$-free graphs has garnered much attention. In this paper, we are interested in finding a linear subfamily of a specific superclass of $2K_2$-free graphs, namely $(P_3\\cup P_2)$-free graphs. We show that the class of $\\{P_3\\cup P_2,gem\\}$-free graphs admits $f=2\\omega$ as a linear $\\chi$-binding function. Furthermore, we give examples to show that the optimal $\\chi$-binding function $f^*\\geq \\left\\lceil\\frac{5\\ome"},"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":"2305.11757","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.CO","submitted_at":"2023-05-19T15:44:14Z","cross_cats_sorted":[],"title_canon_sha256":"10d15d77cf306bd86ff08e0e6fe4f2b785149a617392daca4bf7985e64c198c7","abstract_canon_sha256":"9d0668294e29081398027cf8e9ed1669bd4f5f206c460552ebafe6788da9918c"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T06:11:50.814341Z","signature_b64":"L3j0nzFA1cRBHbikKx4ip4cujtyGNMDo91NhMVD1bf3fylSj/BGSpTfEWOKidQfptFRjDP0uO2s5fYjPoprtAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"09be79b270d5c5a546c77d9dd87548459dc947b225f011f96e42150b289b14fe","last_reissued_at":"2026-07-05T06:11:50.813912Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T06:11:50.813912Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Linear $\\chi$-binding functions for $\\{P_3\\cup P_2, gem\\}$-free graphs","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Athmakoori Prashant, M. Gokulnath, S. Francis Raj","submitted_at":"2023-05-19T15:44:14Z","abstract_excerpt":"Finding families that admit a linear $\\chi$-binding function is a problem that has interested researchers for a long time. Recently, the question of finding linear subfamilies of $2K_2$-free graphs has garnered much attention. In this paper, we are interested in finding a linear subfamily of a specific superclass of $2K_2$-free graphs, namely $(P_3\\cup P_2)$-free graphs. We show that the class of $\\{P_3\\cup P_2,gem\\}$-free graphs admits $f=2\\omega$ as a linear $\\chi$-binding function. Furthermore, we give examples to show that the optimal $\\chi$-binding function $f^*\\geq \\left\\lceil\\frac{5\\ome"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2305.11757","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/2305.11757/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":"2305.11757","created_at":"2026-07-05T06:11:50.813969+00:00"},{"alias_kind":"arxiv_version","alias_value":"2305.11757v1","created_at":"2026-07-05T06:11:50.813969+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2305.11757","created_at":"2026-07-05T06:11:50.813969+00:00"},{"alias_kind":"pith_short_12","alias_value":"BG7HTMTQ2XC2","created_at":"2026-07-05T06:11:50.813969+00:00"},{"alias_kind":"pith_short_16","alias_value":"BG7HTMTQ2XC2KRWH","created_at":"2026-07-05T06:11:50.813969+00:00"},{"alias_kind":"pith_short_8","alias_value":"BG7HTMTQ","created_at":"2026-07-05T06:11:50.813969+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2412.14524","citing_title":"Coloring of some $(P_2\\cup P_4)$-free graphs","ref_index":14,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW","json":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW.json","graph_json":"https://pith.science/api/pith-number/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/graph.json","events_json":"https://pith.science/api/pith-number/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/events.json","paper":"https://pith.science/paper/BG7HTMTQ"},"agent_actions":{"view_html":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW","download_json":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW.json","view_paper":"https://pith.science/paper/BG7HTMTQ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2305.11757&json=true","fetch_graph":"https://pith.science/api/pith-number/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/graph.json","fetch_events":"https://pith.science/api/pith-number/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/action/storage_attestation","attest_author":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/action/author_attestation","sign_citation":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/action/citation_signature","submit_replication":"https://pith.science/pith/BG7HTMTQ2XC2KRWHPWO5Q5KIIW/action/replication_record"}},"created_at":"2026-07-05T06:11:50.813969+00:00","updated_at":"2026-07-05T06:11:50.813969+00:00"}