{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:BWKI4XSUS4NT4UBOMTNEFG6XBN","short_pith_number":"pith:BWKI4XSU","canonical_record":{"source":{"id":"1606.05045","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-16T03:59:56Z","cross_cats_sorted":[],"title_canon_sha256":"51b5df7ca7ffd3e3a99b76b864d3eaafdc1e9236f01dc44790db54034980a802","abstract_canon_sha256":"c4dd7050bc79be971851863633b99cad4e6680a76affc983902af78f980a97a3"},"schema_version":"1.0"},"canonical_sha256":"0d948e5e54971b3e502e64da429bd70b6408b0a79d8e9c6f58fcd1f7995af18a","source":{"kind":"arxiv","id":"1606.05045","version":4},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.05045","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"arxiv_version","alias_value":"1606.05045v4","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05045","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"pith_short_12","alias_value":"BWKI4XSUS4NT","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"BWKI4XSUS4NT4UBO","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"BWKI4XSU","created_at":"2026-05-18T12:30:09Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:BWKI4XSUS4NT4UBOMTNEFG6XBN","target":"record","payload":{"canonical_record":{"source":{"id":"1606.05045","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-16T03:59:56Z","cross_cats_sorted":[],"title_canon_sha256":"51b5df7ca7ffd3e3a99b76b864d3eaafdc1e9236f01dc44790db54034980a802","abstract_canon_sha256":"c4dd7050bc79be971851863633b99cad4e6680a76affc983902af78f980a97a3"},"schema_version":"1.0"},"canonical_sha256":"0d948e5e54971b3e502e64da429bd70b6408b0a79d8e9c6f58fcd1f7995af18a","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:06:24.155768Z","signature_b64":"CX5gqpZCVaCCTnuBGdk6qXyEVE3rX5Gq1zb4WJmMyXuPFrgmjtvnkk/x1PxWjF9iRfW3ug8UyOKkzaquexlGDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0d948e5e54971b3e502e64da429bd70b6408b0a79d8e9c6f58fcd1f7995af18a","last_reissued_at":"2026-05-18T01:06:24.155247Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:06:24.155247Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.05045","source_version":4,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:06:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"aaqqhIFIO9/Cr/C1ebejxOrmeXuDspBAcAWOUoXDFVqkJlk0AwkSMzUdvCrrmULJ1sSDT2D5SmnuDwjKrJaaAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:45:38.389076Z"},"content_sha256":"cde3dce5937be7129391aeb67a8990be76127131469e05fe3b9ca2394f660347","schema_version":"1.0","event_id":"sha256:cde3dce5937be7129391aeb67a8990be76127131469e05fe3b9ca2394f660347"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:BWKI4XSUS4NT4UBOMTNEFG6XBN","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Some properties and applications of odd-colorable $r$-hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chen Ouyang, Jiayu Shao, Liqun Qi, Xiying Yuan","submitted_at":"2016-06-16T03:59:56Z","abstract_excerpt":"Let $r\\geq2$ and $r$ be even. An $r$-hypergraph $G$ on $n$ vertices is called odd-colorable if there exists a map $\\varphi:[n]\\rightarrow\\lbrack r]$ such that for any edge $\\{j_{1},j_{2},\\cdots,j_{r}\\}$ of $G$, we have $\\varphi(j_{1})+\\varphi(j_{2})+\\cdot\\cdot\\cdot+\\varphi(j_{r})\\equiv r/2(\\operatorname{mod}r).$ In this paper, we first determine that, if $r=2^{q}(2t+1)$ and $n\\ge 2^{q}(2^{q}-1)r$, then the maximum chromatic number in the class of the odd-colorable $r$-hypergraphs on $n$ vertices is $2^q$, which answers a question raised by V. Nikiforov recently in [V. Nikiforov, Hypergraphs an"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05045","kind":"arxiv","version":4},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:06:24Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1gjvOGlfXEBCFqjegi3CTDLheU2exSjz3OXMAvfT98qm/5IZZlAsOn3ai9ePQGOTeBISbAsYMkMlq2Wt2/mSDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-03T07:45:38.389438Z"},"content_sha256":"5778374b80cdf4ec932b8fc1d7bcfedadc30c0db1353586858e18b451be2d2bd","schema_version":"1.0","event_id":"sha256:5778374b80cdf4ec932b8fc1d7bcfedadc30c0db1353586858e18b451be2d2bd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BWKI4XSUS4NT4UBOMTNEFG6XBN/bundle.json","state_url":"https://pith.science/pith/BWKI4XSUS4NT4UBOMTNEFG6XBN/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BWKI4XSUS4NT4UBOMTNEFG6XBN/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-03T07:45:38Z","links":{"resolver":"https://pith.science/pith/BWKI4XSUS4NT4UBOMTNEFG6XBN","bundle":"https://pith.science/pith/BWKI4XSUS4NT4UBOMTNEFG6XBN/bundle.json","state":"https://pith.science/pith/BWKI4XSUS4NT4UBOMTNEFG6XBN/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BWKI4XSUS4NT4UBOMTNEFG6XBN/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:BWKI4XSUS4NT4UBOMTNEFG6XBN","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":"c4dd7050bc79be971851863633b99cad4e6680a76affc983902af78f980a97a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-16T03:59:56Z","title_canon_sha256":"51b5df7ca7ffd3e3a99b76b864d3eaafdc1e9236f01dc44790db54034980a802"},"schema_version":"1.0","source":{"id":"1606.05045","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.05045","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"arxiv_version","alias_value":"1606.05045v4","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.05045","created_at":"2026-05-18T01:06:24Z"},{"alias_kind":"pith_short_12","alias_value":"BWKI4XSUS4NT","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_16","alias_value":"BWKI4XSUS4NT4UBO","created_at":"2026-05-18T12:30:09Z"},{"alias_kind":"pith_short_8","alias_value":"BWKI4XSU","created_at":"2026-05-18T12:30:09Z"}],"graph_snapshots":[{"event_id":"sha256:5778374b80cdf4ec932b8fc1d7bcfedadc30c0db1353586858e18b451be2d2bd","target":"graph","created_at":"2026-05-18T01:06:24Z","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"},"paper":{"abstract_excerpt":"Let $r\\geq2$ and $r$ be even. An $r$-hypergraph $G$ on $n$ vertices is called odd-colorable if there exists a map $\\varphi:[n]\\rightarrow\\lbrack r]$ such that for any edge $\\{j_{1},j_{2},\\cdots,j_{r}\\}$ of $G$, we have $\\varphi(j_{1})+\\varphi(j_{2})+\\cdot\\cdot\\cdot+\\varphi(j_{r})\\equiv r/2(\\operatorname{mod}r).$ In this paper, we first determine that, if $r=2^{q}(2t+1)$ and $n\\ge 2^{q}(2^{q}-1)r$, then the maximum chromatic number in the class of the odd-colorable $r$-hypergraphs on $n$ vertices is $2^q$, which answers a question raised by V. Nikiforov recently in [V. Nikiforov, Hypergraphs an","authors_text":"Chen Ouyang, Jiayu Shao, Liqun Qi, Xiying Yuan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-16T03:59:56Z","title":"Some properties and applications of odd-colorable $r$-hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.05045","kind":"arxiv","version":4},"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:cde3dce5937be7129391aeb67a8990be76127131469e05fe3b9ca2394f660347","target":"record","created_at":"2026-05-18T01:06:24Z","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":"c4dd7050bc79be971851863633b99cad4e6680a76affc983902af78f980a97a3","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-06-16T03:59:56Z","title_canon_sha256":"51b5df7ca7ffd3e3a99b76b864d3eaafdc1e9236f01dc44790db54034980a802"},"schema_version":"1.0","source":{"id":"1606.05045","kind":"arxiv","version":4}},"canonical_sha256":"0d948e5e54971b3e502e64da429bd70b6408b0a79d8e9c6f58fcd1f7995af18a","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0d948e5e54971b3e502e64da429bd70b6408b0a79d8e9c6f58fcd1f7995af18a","first_computed_at":"2026-05-18T01:06:24.155247Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:06:24.155247Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CX5gqpZCVaCCTnuBGdk6qXyEVE3rX5Gq1zb4WJmMyXuPFrgmjtvnkk/x1PxWjF9iRfW3ug8UyOKkzaquexlGDA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:06:24.155768Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.05045","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:cde3dce5937be7129391aeb67a8990be76127131469e05fe3b9ca2394f660347","sha256:5778374b80cdf4ec932b8fc1d7bcfedadc30c0db1353586858e18b451be2d2bd"],"state_sha256":"bc89a5365b87d7857c11c593ca31eb8f2b62b2213cf653894c1d768e8e6b6176"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0wLp7AdBAQ/jyfTV8EceoAQsZ19sApyGFT8hweBiK4jh1YFeD9hTSAyRsLKtgRt0kC5yYFr4y+xF5Pk8oIByBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-03T07:45:38.391420Z","bundle_sha256":"ca3e53fb4dcbdb96b9effe3b38191cc4b6e71086d5cd025819d93356c0a36e43"}}