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Chen, Xu and Xu conjectured that if $q\\ge2$ and $G$ is a $(\\mathrm{cap},\\mathrm{even\\ hole})$-free graph with no odd hole of length at most $2q-1$, then $\\chi(G)\\le \\left\\lceil \\frac{2q+1}{2q}\\omega(G)\\right\\rceil.$\n  They confirmed the conjecture for $q \\le 3$. In this paper, we prove the conjecture for all $q \\ge 3$. As a corollary, we prove that for such a graph $G$, $\\chi_f(G)\\le \\frac{2q+1}{2q}\\omega(G).$"},"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":"2607.27850","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-30T08:29:46Z","cross_cats_sorted":[],"title_canon_sha256":"6cc5f62a14ff390f70a4490153c39cf89050b20fbfa202fd1a4b3331b7bc3ba9","abstract_canon_sha256":"1541edf2ce9a26b09ee13d3b178687490b439302b22878319d401350a27d3486"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4699c7fab7003ab2e5c56973fdee24f3d9f2b627e4dd2049d7eaaaf49e08cc0a","last_reissued_at":"2026-07-31T01:34:28.734894Z","signature_status":"unsigned_v0","first_computed_at":"2026-07-31T01:34:28.734894Z"},"graph_snapshot":{"paper":{"title":"Optimal binding function for (cap,even hole)-free graphs with no short odd holes","license":"http://creativecommons.org/publicdomain/zero/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chenglong Deng, Xuding Zhu","submitted_at":"2026-07-30T08:29:46Z","abstract_excerpt":"A hole in a graph is an induced cycle of length at least $4$. A cap is a hole together with a vertex adjacent to exactly two consecutive vertices of it. Chen, Xu and Xu conjectured that if $q\\ge2$ and $G$ is a $(\\mathrm{cap},\\mathrm{even\\ hole})$-free graph with no odd hole of length at most $2q-1$, then $\\chi(G)\\le \\left\\lceil \\frac{2q+1}{2q}\\omega(G)\\right\\rceil.$\n  They confirmed the conjecture for $q \\le 3$. In this paper, we prove the conjecture for all $q \\ge 3$. As a corollary, we prove that for such a graph $G$, $\\chi_f(G)\\le \\frac{2q+1}{2q}\\omega(G).$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.27850","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/2607.27850/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":"2607.27850","created_at":"2026-07-31T01:34:28.737953+00:00"},{"alias_kind":"arxiv_version","alias_value":"2607.27850v1","created_at":"2026-07-31T01:34:28.737953+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.27850","created_at":"2026-07-31T01:34:28.737953+00:00"},{"alias_kind":"pith_short_12","alias_value":"I2M4P6VXAA5L","created_at":"2026-07-31T01:34:28.737953+00:00"},{"alias_kind":"pith_short_16","alias_value":"I2M4P6VXAA5LFZOF","created_at":"2026-07-31T01:34:28.737953+00:00"},{"alias_kind":"pith_short_8","alias_value":"I2M4P6VX","created_at":"2026-07-31T01:34:28.737953+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/I2M4P6VXAA5LFZOFNFZ733RE6P","json":"https://pith.science/pith/I2M4P6VXAA5LFZOFNFZ733RE6P.json","graph_json":"https://pith.science/api/pith-number/I2M4P6VXAA5LFZOFNFZ733RE6P/graph.json","events_json":"https://pith.science/api/pith-number/I2M4P6VXAA5LFZOFNFZ733RE6P/events.json","paper":"https://pith.science/paper/I2M4P6VX"},"agent_actions":{"view_html":"https://pith.science/pith/I2M4P6VXAA5LFZOFNFZ733RE6P","download_json":"https://pith.science/pith/I2M4P6VXAA5LFZOFNFZ733RE6P.json","view_paper":"https://pith.science/paper/I2M4P6VX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2607.27850&json=true","fetch_graph":"https://pith.science/api/pith-number/I2M4P6VXAA5LFZOFNFZ733RE6P/graph.json","fetch_events":"https://pith.science/api/pith-number/I2M4P6VXAA5LFZOFNFZ733RE6P/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/I2M4P6VXAA5LFZOFNFZ733RE6P/action/timestamp_anchor","attest_storage":"https://pith.science/pith/I2M4P6VXAA5LFZOFNFZ733RE6P/action/storage_attestation","attest_author":"https://pith.science/pith/I2M4P6VXAA5LFZOFNFZ733RE6P/action/author_attestation","sign_citation":"https://pith.science/pith/I2M4P6VXAA5LFZOFNFZ733RE6P/action/citation_signature","submit_replication":"https://pith.science/pith/I2M4P6VXAA5LFZOFNFZ733RE6P/action/replication_record"}},"created_at":"2026-07-31T01:34:28.737953+00:00","updated_at":"2026-07-31T01:34:28.737953+00:00"}