{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:G7KO3FFB72YDKPDKLRBDMPYK7I","short_pith_number":"pith:G7KO3FFB","canonical_record":{"source":{"id":"1209.3646","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-17T13:25:36Z","cross_cats_sorted":[],"title_canon_sha256":"a85a920e4cde7ee7a5898d6051aa015b717981206d6a4fac1223fa7d22822f4a","abstract_canon_sha256":"5a246ca5923fe220fcc61b403f2ad0ce201c698b55c46a52b9c2198d7ea4865b"},"schema_version":"1.0"},"canonical_sha256":"37d4ed94a1feb0353c6a5c42363f0afa09a971a7e404009559800dc9d6c43c58","source":{"kind":"arxiv","id":"1209.3646","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3646","created_at":"2026-05-18T03:44:23Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3646v3","created_at":"2026-05-18T03:44:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3646","created_at":"2026-05-18T03:44:23Z"},{"alias_kind":"pith_short_12","alias_value":"G7KO3FFB72YD","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"G7KO3FFB72YDKPDK","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"G7KO3FFB","created_at":"2026-05-18T12:27:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:G7KO3FFB72YDKPDKLRBDMPYK7I","target":"record","payload":{"canonical_record":{"source":{"id":"1209.3646","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-17T13:25:36Z","cross_cats_sorted":[],"title_canon_sha256":"a85a920e4cde7ee7a5898d6051aa015b717981206d6a4fac1223fa7d22822f4a","abstract_canon_sha256":"5a246ca5923fe220fcc61b403f2ad0ce201c698b55c46a52b9c2198d7ea4865b"},"schema_version":"1.0"},"canonical_sha256":"37d4ed94a1feb0353c6a5c42363f0afa09a971a7e404009559800dc9d6c43c58","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:44:23.619495Z","signature_b64":"nraevcwYjxLgv9Y/oE4Q6qrGavEhvyyfashPqYfHO2SWANjMo5opTukkdfXV2zQ3890tNZcrBkDg6mP54011Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"37d4ed94a1feb0353c6a5c42363f0afa09a971a7e404009559800dc9d6c43c58","last_reissued_at":"2026-05-18T03:44:23.618979Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:44:23.618979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.3646","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T03:44:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Gh79wb4ov40qTEPAkrJvdu2rR/FcnKcuTAAR/SHtBnThLtJU8OrcVd2t3YDkSis3vrxlEKzSSJwXR8gZcpE/Cw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:34:42.678480Z"},"content_sha256":"0c99c0f9c52e5a4e07da140efa76290341e8432cca80bbbe0832c059daa8221b","schema_version":"1.0","event_id":"sha256:0c99c0f9c52e5a4e07da140efa76290341e8432cca80bbbe0832c059daa8221b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:G7KO3FFB72YDKPDKLRBDMPYK7I","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Coloring graphs with dense neighborhoods","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Landon Rabern","submitted_at":"2012-09-17T13:25:36Z","abstract_excerpt":"It is shown that any graph with maximum degree $\\Delta$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\\frac{6k^2}{6k^2 + 1}\\Delta + k + 6$ is either $(\\Delta - k)$-colorable or contains a clique on more than $\\Delta - 2k$ vertices. In the $k=1$ case we improve the bound on the average degree to $\\frac23\\Delta + 4$ and the bound on the clique number to $\\Delta-1$. As corollaries, we show that every graph satisfies $\\chi \\leq \\max\\set{\\omega, \\Delta - 1, 4\\alpha}$ and every graph satisfies $\\chi \\leq \\max\\set{\\omega, \\Delta - 1, \\ceil{\\fra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3646","kind":"arxiv","version":3},"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":"2026-05-18T03:44:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CidZMljo65Y0AMo5J42fSOC9oPT9MLfO131SkYkTYGp8ECtLNO+iqQbAAdJnXRABYJjJs1MaRYlZ5k8+R2dADQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-10T23:34:42.679504Z"},"content_sha256":"524c56ad7bb47d0e2283d0ecd4428937fd3f0a7a75f9cbb13001a0db869b5286","schema_version":"1.0","event_id":"sha256:524c56ad7bb47d0e2283d0ecd4428937fd3f0a7a75f9cbb13001a0db869b5286"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/G7KO3FFB72YDKPDKLRBDMPYK7I/bundle.json","state_url":"https://pith.science/pith/G7KO3FFB72YDKPDKLRBDMPYK7I/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/G7KO3FFB72YDKPDKLRBDMPYK7I/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":"2026-06-10T23:34:42Z","links":{"resolver":"https://pith.science/pith/G7KO3FFB72YDKPDKLRBDMPYK7I","bundle":"https://pith.science/pith/G7KO3FFB72YDKPDKLRBDMPYK7I/bundle.json","state":"https://pith.science/pith/G7KO3FFB72YDKPDKLRBDMPYK7I/state.json","well_known_bundle":"https://pith.science/.well-known/pith/G7KO3FFB72YDKPDKLRBDMPYK7I/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:G7KO3FFB72YDKPDKLRBDMPYK7I","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":"5a246ca5923fe220fcc61b403f2ad0ce201c698b55c46a52b9c2198d7ea4865b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-17T13:25:36Z","title_canon_sha256":"a85a920e4cde7ee7a5898d6051aa015b717981206d6a4fac1223fa7d22822f4a"},"schema_version":"1.0","source":{"id":"1209.3646","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.3646","created_at":"2026-05-18T03:44:23Z"},{"alias_kind":"arxiv_version","alias_value":"1209.3646v3","created_at":"2026-05-18T03:44:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.3646","created_at":"2026-05-18T03:44:23Z"},{"alias_kind":"pith_short_12","alias_value":"G7KO3FFB72YD","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_16","alias_value":"G7KO3FFB72YDKPDK","created_at":"2026-05-18T12:27:06Z"},{"alias_kind":"pith_short_8","alias_value":"G7KO3FFB","created_at":"2026-05-18T12:27:06Z"}],"graph_snapshots":[{"event_id":"sha256:524c56ad7bb47d0e2283d0ecd4428937fd3f0a7a75f9cbb13001a0db869b5286","target":"graph","created_at":"2026-05-18T03:44:23Z","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":"It is shown that any graph with maximum degree $\\Delta$ in which the average degree of the induced subgraph on the set of all neighbors of any vertex exceeds $\\frac{6k^2}{6k^2 + 1}\\Delta + k + 6$ is either $(\\Delta - k)$-colorable or contains a clique on more than $\\Delta - 2k$ vertices. In the $k=1$ case we improve the bound on the average degree to $\\frac23\\Delta + 4$ and the bound on the clique number to $\\Delta-1$. As corollaries, we show that every graph satisfies $\\chi \\leq \\max\\set{\\omega, \\Delta - 1, 4\\alpha}$ and every graph satisfies $\\chi \\leq \\max\\set{\\omega, \\Delta - 1, \\ceil{\\fra","authors_text":"Landon Rabern","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-17T13:25:36Z","title":"Coloring graphs with dense neighborhoods"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.3646","kind":"arxiv","version":3},"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":"sha256:0c99c0f9c52e5a4e07da140efa76290341e8432cca80bbbe0832c059daa8221b","target":"record","created_at":"2026-05-18T03:44:23Z","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":"5a246ca5923fe220fcc61b403f2ad0ce201c698b55c46a52b9c2198d7ea4865b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-17T13:25:36Z","title_canon_sha256":"a85a920e4cde7ee7a5898d6051aa015b717981206d6a4fac1223fa7d22822f4a"},"schema_version":"1.0","source":{"id":"1209.3646","kind":"arxiv","version":3}},"canonical_sha256":"37d4ed94a1feb0353c6a5c42363f0afa09a971a7e404009559800dc9d6c43c58","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"37d4ed94a1feb0353c6a5c42363f0afa09a971a7e404009559800dc9d6c43c58","first_computed_at":"2026-05-18T03:44:23.618979Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:44:23.618979Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"nraevcwYjxLgv9Y/oE4Q6qrGavEhvyyfashPqYfHO2SWANjMo5opTukkdfXV2zQ3890tNZcrBkDg6mP54011Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:44:23.619495Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.3646","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0c99c0f9c52e5a4e07da140efa76290341e8432cca80bbbe0832c059daa8221b","sha256:524c56ad7bb47d0e2283d0ecd4428937fd3f0a7a75f9cbb13001a0db869b5286"],"state_sha256":"fb04519ec676117a811b7ebd8ae1e253485b5f6e7c72466c3cde8ad883bbe130"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"VnrT82BAWB8u+4U87ECeGc1lu5+xfIHuNxqDcmJuSwS3FmoNgngI3l+8XANvmAOktBhfDGq7o5ZUJlU99kNzDA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-10T23:34:42.683124Z","bundle_sha256":"8d2c059b2124a9f2c9b4e48ca363a421307a0d6ce8e3b9d616d60d01c8d2aa2d"}}