{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:BOT3QKTK5L7HNWK4CQQ26WLATP","short_pith_number":"pith:BOT3QKTK","canonical_record":{"source":{"id":"1109.2112","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-09-09T19:58:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1569f83618ae22189cdce271c02081b5b08a976bf5d5d7873848c6fb5b4a062f","abstract_canon_sha256":"0fd3bca41e0e94f7390ca4383f271531ba34741141625af899b2c644905fb5c7"},"schema_version":"1.0"},"canonical_sha256":"0ba7b82a6aeafe76d95c1421af59609bd5c0aacc04026c7aa16d85764d807506","source":{"kind":"arxiv","id":"1109.2112","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2112","created_at":"2026-05-18T04:07:29Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2112v2","created_at":"2026-05-18T04:07:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2112","created_at":"2026-05-18T04:07:29Z"},{"alias_kind":"pith_short_12","alias_value":"BOT3QKTK5L7H","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BOT3QKTK5L7HNWK4","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BOT3QKTK","created_at":"2026-05-18T12:26:24Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:BOT3QKTK5L7HNWK4CQQ26WLATP","target":"record","payload":{"canonical_record":{"source":{"id":"1109.2112","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-09-09T19:58:34Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"1569f83618ae22189cdce271c02081b5b08a976bf5d5d7873848c6fb5b4a062f","abstract_canon_sha256":"0fd3bca41e0e94f7390ca4383f271531ba34741141625af899b2c644905fb5c7"},"schema_version":"1.0"},"canonical_sha256":"0ba7b82a6aeafe76d95c1421af59609bd5c0aacc04026c7aa16d85764d807506","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:07:29.013030Z","signature_b64":"LdtiR+R2JJ3urtNihtJzzF+8hmx2ZxbGLIajzKxYKdtc/99/2HmUD//eg4MBokxyR0lrxRz08VHRNfBdjckYDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0ba7b82a6aeafe76d95c1421af59609bd5c0aacc04026c7aa16d85764d807506","last_reissued_at":"2026-05-18T04:07:29.012609Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:07:29.012609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.2112","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:07:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ab7lP/jSEJD6AJRwLQdRR9Xm3O+ELfp5ZVyiQRQtL29cLDrUkvxBos+TrMSVjXj0JbzRzu84viuDQpRKlR37DQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T04:43:59.945151Z"},"content_sha256":"16c5b487d8c882e39aac5e85f44cd090707bec8efd37386de50c177d53b00e9a","schema_version":"1.0","event_id":"sha256:16c5b487d8c882e39aac5e85f44cd090707bec8efd37386de50c177d53b00e9a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:BOT3QKTK5L7HNWK4CQQ26WLATP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"A local strengthening of Reed's {\\omega}, \\Delta, {\\chi} conjecture for quasi-line graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Andrew D. King, Maria Chudnovsky, Matthieu Plumettaz, Paul Seymour","submitted_at":"2011-09-09T19:58:34Z","abstract_excerpt":"Reed's $\\omega$, $\\Delta$, $\\chi$ conjecture proposes that every graph satisfies $\\chi\\leq \\lceil\\frac 12(\\Delta+1+\\omega)\\rceil$; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and $O(n^3m^2)$ for quasi-line graphs. For line graphs, this is faster tha"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2112","kind":"arxiv","version":2},"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":"2026-05-18T04:07:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"psxmn4ww5hphPbhM3x2wYYdU40hc+jppoOyhxqNGe62YvirQ6H++lgTRlZjtz8mAn8c/76N5u9IPk9Q/cup9AA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-28T04:43:59.945514Z"},"content_sha256":"d1e56a0c93b436f6c645f4c5589056371957c05296d07f4b9613b0f0aa359369","schema_version":"1.0","event_id":"sha256:d1e56a0c93b436f6c645f4c5589056371957c05296d07f4b9613b0f0aa359369"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/BOT3QKTK5L7HNWK4CQQ26WLATP/bundle.json","state_url":"https://pith.science/pith/BOT3QKTK5L7HNWK4CQQ26WLATP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/BOT3QKTK5L7HNWK4CQQ26WLATP/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":"2026-05-28T04:43:59Z","links":{"resolver":"https://pith.science/pith/BOT3QKTK5L7HNWK4CQQ26WLATP","bundle":"https://pith.science/pith/BOT3QKTK5L7HNWK4CQQ26WLATP/bundle.json","state":"https://pith.science/pith/BOT3QKTK5L7HNWK4CQQ26WLATP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/BOT3QKTK5L7HNWK4CQQ26WLATP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:BOT3QKTK5L7HNWK4CQQ26WLATP","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":"0fd3bca41e0e94f7390ca4383f271531ba34741141625af899b2c644905fb5c7","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-09-09T19:58:34Z","title_canon_sha256":"1569f83618ae22189cdce271c02081b5b08a976bf5d5d7873848c6fb5b4a062f"},"schema_version":"1.0","source":{"id":"1109.2112","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2112","created_at":"2026-05-18T04:07:29Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2112v2","created_at":"2026-05-18T04:07:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2112","created_at":"2026-05-18T04:07:29Z"},{"alias_kind":"pith_short_12","alias_value":"BOT3QKTK5L7H","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_16","alias_value":"BOT3QKTK5L7HNWK4","created_at":"2026-05-18T12:26:24Z"},{"alias_kind":"pith_short_8","alias_value":"BOT3QKTK","created_at":"2026-05-18T12:26:24Z"}],"graph_snapshots":[{"event_id":"sha256:d1e56a0c93b436f6c645f4c5589056371957c05296d07f4b9613b0f0aa359369","target":"graph","created_at":"2026-05-18T04:07:29Z","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":"Reed's $\\omega$, $\\Delta$, $\\chi$ conjecture proposes that every graph satisfies $\\chi\\leq \\lceil\\frac 12(\\Delta+1+\\omega)\\rceil$; it is known to hold for all claw-free graphs. In this paper we consider a local strengthening of this conjecture. We prove the local strengthening for line graphs, then note that previous results immediately tell us that the local strengthening holds for all quasi-line graphs. Our proofs lead to polytime algorithms for constructing colourings that achieve our bounds: $O(n^2)$ for line graphs and $O(n^3m^2)$ for quasi-line graphs. For line graphs, this is faster tha","authors_text":"Andrew D. King, Maria Chudnovsky, Matthieu Plumettaz, Paul Seymour","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-09-09T19:58:34Z","title":"A local strengthening of Reed's {\\omega}, \\Delta, {\\chi} conjecture for quasi-line graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2112","kind":"arxiv","version":2},"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":"sha256:16c5b487d8c882e39aac5e85f44cd090707bec8efd37386de50c177d53b00e9a","target":"record","created_at":"2026-05-18T04:07:29Z","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":"0fd3bca41e0e94f7390ca4383f271531ba34741141625af899b2c644905fb5c7","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2011-09-09T19:58:34Z","title_canon_sha256":"1569f83618ae22189cdce271c02081b5b08a976bf5d5d7873848c6fb5b4a062f"},"schema_version":"1.0","source":{"id":"1109.2112","kind":"arxiv","version":2}},"canonical_sha256":"0ba7b82a6aeafe76d95c1421af59609bd5c0aacc04026c7aa16d85764d807506","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"0ba7b82a6aeafe76d95c1421af59609bd5c0aacc04026c7aa16d85764d807506","first_computed_at":"2026-05-18T04:07:29.012609Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:07:29.012609Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LdtiR+R2JJ3urtNihtJzzF+8hmx2ZxbGLIajzKxYKdtc/99/2HmUD//eg4MBokxyR0lrxRz08VHRNfBdjckYDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:07:29.013030Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.2112","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:16c5b487d8c882e39aac5e85f44cd090707bec8efd37386de50c177d53b00e9a","sha256:d1e56a0c93b436f6c645f4c5589056371957c05296d07f4b9613b0f0aa359369"],"state_sha256":"f042f4183b7911d567477222319b1251f6ec538daaf7463272d1069aeeeed53e"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ZJwiYVqo/nX7gGpCQwNLxVKdGVpgE2Us7KNN5jUC0Q5JnVZiDI0tf9sBBHtFGOnxf8jHXEr1uffpe3T20OW6Cw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-28T04:43:59.947482Z","bundle_sha256":"dd747beee7a08d9487731a6a3e12aa0bd20925cbeb672c0a641c020761d56a29"}}