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In this paper, we prove that the perfect divisibility of fork-free graphs is equivalent to that of claw-free graphs. We also prove that, for $F\\in \\{P_7, P_6\\cup K_1\\}$, each (fork, $F$)-free graph $G$ is perfectly divisible and hence $\\chi(G)\\leq \\binom{\\omega(G)+1}{2}$."},"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":"2504.14863","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2025-04-21T05:08:16Z","cross_cats_sorted":[],"title_canon_sha256":"fc587ed95194a753fc7a46aa61abb49f9f98b159ee9104b1b7d29bb3ec78c688","abstract_canon_sha256":"78149f3d243cdb6392cbb129c6a47de7da08ab6195467014a6e29bc8742980b9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:52:23.309589Z","signature_b64":"Jw5t8P/FasGhnaSVmggImuojmREaLkvZXPqjUUCk7dRbcemUVKELbW4+lM3IM92QLga0GvNHro5cHI36QYayAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2df940ab5884fc8ecd619f824cd8c780f687787503cec939b18e833507a91042","last_reissued_at":"2026-07-05T10:52:23.309080Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:52:23.309080Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On minimal nonperfectly divisible fork-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Baogang Xu, Miaoxia Zhuang","submitted_at":"2025-04-21T05:08:16Z","abstract_excerpt":"A fork is a graph obtained from $K_{1,3}$ (usually called claw) by subdividing an edge once. A graph is perfectly divisible if for each of its induced subgraph $H$, $V(H)$ can be partitioned into $A$ and $B$ such that $H[A]$ is perfect and $\\omega(H[B]) < \\omega(H)$. In this paper, we prove that the perfect divisibility of fork-free graphs is equivalent to that of claw-free graphs. We also prove that, for $F\\in \\{P_7, P_6\\cup K_1\\}$, each (fork, $F$)-free graph $G$ is perfectly divisible and hence $\\chi(G)\\leq \\binom{\\omega(G)+1}{2}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2504.14863","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/2504.14863/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":"2504.14863","created_at":"2026-07-05T10:52:23.309139+00:00"},{"alias_kind":"arxiv_version","alias_value":"2504.14863v2","created_at":"2026-07-05T10:52:23.309139+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2504.14863","created_at":"2026-07-05T10:52:23.309139+00:00"},{"alias_kind":"pith_short_12","alias_value":"FX4UBK2YQT6I","created_at":"2026-07-05T10:52:23.309139+00:00"},{"alias_kind":"pith_short_16","alias_value":"FX4UBK2YQT6I5TLB","created_at":"2026-07-05T10:52:23.309139+00:00"},{"alias_kind":"pith_short_8","alias_value":"FX4UBK2Y","created_at":"2026-07-05T10:52:23.309139+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2608.13519","citing_title":"Every fork-free graph is perfectly weight divisible","ref_index":30,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD","json":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD.json","graph_json":"https://pith.science/api/pith-number/FX4UBK2YQT6I5TLBT6BEZWGHQD/graph.json","events_json":"https://pith.science/api/pith-number/FX4UBK2YQT6I5TLBT6BEZWGHQD/events.json","paper":"https://pith.science/paper/FX4UBK2Y"},"agent_actions":{"view_html":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD","download_json":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD.json","view_paper":"https://pith.science/paper/FX4UBK2Y","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2504.14863&json=true","fetch_graph":"https://pith.science/api/pith-number/FX4UBK2YQT6I5TLBT6BEZWGHQD/graph.json","fetch_events":"https://pith.science/api/pith-number/FX4UBK2YQT6I5TLBT6BEZWGHQD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD/action/storage_attestation","attest_author":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD/action/author_attestation","sign_citation":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD/action/citation_signature","submit_replication":"https://pith.science/pith/FX4UBK2YQT6I5TLBT6BEZWGHQD/action/replication_record"}},"created_at":"2026-07-05T10:52:23.309139+00:00","updated_at":"2026-07-05T10:52:23.309139+00:00"}