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A {\\em bull} consists of a triangle with two disjoint pendant edges, a {\\em hammer} is obtained by identifying an end of $P_3$ with a vertex of a triangle, a {\\em fork$^+$} is obtained from $K_{1, 3}$ by subdividing an edge twice. Let $H$ be a bull or a hammer, and $F$ be a $P_7$ or a fork$^+$. We determine all $(C_"},"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":"2409.06944","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-09-11T01:55:23Z","cross_cats_sorted":[],"title_canon_sha256":"528c4b5dbca048bdcc58f2301e7483e233d2ede90318555fa4eed183257acf14","abstract_canon_sha256":"e56399f25802b2d7838d2f43f0991e200b0c1c2ea6f47454608dd4fab5ff812b"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:05:43.475439Z","signature_b64":"ruaZCWNz5asbD7601471h4RQtP3tcxpHNmRJVzgErUncYDMhFtl3AEzEO0nzhmOXFovumMTMMAitDOWPlFGqDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"717902fa35ee7cd0e20636d0ae2a1ed7af1f97140f2e71346233f205e23368a2","last_reissued_at":"2026-07-05T09:05:43.474950Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:05:43.474950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Nearly optimal coloring of some C4-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Baogang Xu, Ran Chen","submitted_at":"2024-09-11T01:55:23Z","abstract_excerpt":"A class ${\\cal G}$ of graphs is $\\chi$-{\\em polydet} if ${\\cal G}$ has a polynomial binding function $f$ and there is a polynomial time algorithm to determine an $f(\\omega(G))$-coloring of $G\\in {\\cal G}$. Let $P_t$ and $C_t$ denote a path and a cycle on $t$ vertices, respectively. A {\\em bull} consists of a triangle with two disjoint pendant edges, a {\\em hammer} is obtained by identifying an end of $P_3$ with a vertex of a triangle, a {\\em fork$^+$} is obtained from $K_{1, 3}$ by subdividing an edge twice. Let $H$ be a bull or a hammer, and $F$ be a $P_7$ or a fork$^+$. We determine all $(C_"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2409.06944","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/2409.06944/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":"2409.06944","created_at":"2026-07-05T09:05:43.475009+00:00"},{"alias_kind":"arxiv_version","alias_value":"2409.06944v1","created_at":"2026-07-05T09:05:43.475009+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2409.06944","created_at":"2026-07-05T09:05:43.475009+00:00"},{"alias_kind":"pith_short_12","alias_value":"OF4QF6RV5Z6N","created_at":"2026-07-05T09:05:43.475009+00:00"},{"alias_kind":"pith_short_16","alias_value":"OF4QF6RV5Z6NBYQG","created_at":"2026-07-05T09:05:43.475009+00:00"},{"alias_kind":"pith_short_8","alias_value":"OF4QF6RV","created_at":"2026-07-05T09:05:43.475009+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2506.19580","citing_title":"The optimal binding function for (cap, even hole)-free graphs","ref_index":4,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626","json":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626.json","graph_json":"https://pith.science/api/pith-number/OF4QF6RV5Z6NBYQGG3IK4KQ626/graph.json","events_json":"https://pith.science/api/pith-number/OF4QF6RV5Z6NBYQGG3IK4KQ626/events.json","paper":"https://pith.science/paper/OF4QF6RV"},"agent_actions":{"view_html":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626","download_json":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626.json","view_paper":"https://pith.science/paper/OF4QF6RV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2409.06944&json=true","fetch_graph":"https://pith.science/api/pith-number/OF4QF6RV5Z6NBYQGG3IK4KQ626/graph.json","fetch_events":"https://pith.science/api/pith-number/OF4QF6RV5Z6NBYQGG3IK4KQ626/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626/action/storage_attestation","attest_author":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626/action/author_attestation","sign_citation":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626/action/citation_signature","submit_replication":"https://pith.science/pith/OF4QF6RV5Z6NBYQGG3IK4KQ626/action/replication_record"}},"created_at":"2026-07-05T09:05:43.475009+00:00","updated_at":"2026-07-05T09:05:43.475009+00:00"}