{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:I2M4P6VXAA5LFZOFNFZ733RE6P","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"1541edf2ce9a26b09ee13d3b178687490b439302b22878319d401350a27d3486","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-30T08:29:46Z","title_canon_sha256":"6cc5f62a14ff390f70a4490153c39cf89050b20fbfa202fd1a4b3331b7bc3ba9"},"schema_version":"1.0","source":{"id":"2607.27850","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.27850","created_at":"2026-07-31T01:34:28Z"},{"alias_kind":"arxiv_version","alias_value":"2607.27850v1","created_at":"2026-07-31T01:34:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.27850","created_at":"2026-07-31T01:34:28Z"},{"alias_kind":"pith_short_12","alias_value":"I2M4P6VXAA5L","created_at":"2026-07-31T01:34:28Z"},{"alias_kind":"pith_short_16","alias_value":"I2M4P6VXAA5LFZOF","created_at":"2026-07-31T01:34:28Z"},{"alias_kind":"pith_short_8","alias_value":"I2M4P6VX","created_at":"2026-07-31T01:34:28Z"}],"graph_snapshots":[{"event_id":"sha256:796e5188c9622b36489c16bfa645d58819d4feac026f077fd9965d75ee52c918","target":"graph","created_at":"2026-07-31T01:34:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2607.27850/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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 $G$, $\\chi_f(G)\\le \\frac{2q+1}{2q}\\omega(G).$","authors_text":"Chenglong Deng, Xuding Zhu","cross_cats":[],"headline":"","license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-30T08:29:46Z","title":"Optimal binding function for (cap,even hole)-free graphs with no short odd holes"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.27850","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:c256878494327f96825ad614b26647f0341fc669690b2bc5598e39386584c29c","target":"record","created_at":"2026-07-31T01:34:28Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"1541edf2ce9a26b09ee13d3b178687490b439302b22878319d401350a27d3486","cross_cats_sorted":[],"license":"http://creativecommons.org/publicdomain/zero/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-30T08:29:46Z","title_canon_sha256":"6cc5f62a14ff390f70a4490153c39cf89050b20fbfa202fd1a4b3331b7bc3ba9"},"schema_version":"1.0","source":{"id":"2607.27850","kind":"arxiv","version":1}},"canonical_sha256":"4699c7fab7003ab2e5c56973fdee24f3d9f2b627e4dd2049d7eaaaf49e08cc0a","receipt":{"builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"4699c7fab7003ab2e5c56973fdee24f3d9f2b627e4dd2049d7eaaaf49e08cc0a","first_computed_at":"2026-07-31T01:34:28.734894Z","kind":"pith_receipt","last_reissued_at":"2026-07-31T01:34:28.734894Z","receipt_version":"0.3","signature_status":"unsigned_v0"},"source_id":"2607.27850","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:c256878494327f96825ad614b26647f0341fc669690b2bc5598e39386584c29c","sha256:796e5188c9622b36489c16bfa645d58819d4feac026f077fd9965d75ee52c918"],"state_sha256":"94d4963ce81bb6de1f19f8d262ddfa98ff5f6a6f0a81ecc777f080e1742ef9e5"}