{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:CRZY47XJ4HHVEB4T6C56G44PWZ","short_pith_number":"pith:CRZY47XJ","canonical_record":{"source":{"id":"1206.2384","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-06-11T21:33:46Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6e653b6660b14d4c5c76dac02be79ce9b5efb832612a3dd2de077abcb4a845f3","abstract_canon_sha256":"7dafad20b108133f126a061ae9ab054dffa58fcd1d65d0dc0622d4035ed186ab"},"schema_version":"1.0"},"canonical_sha256":"14738e7ee9e1cf520793f0bbe3738fb64dbe7bf614da88dce2e630cb85d82b86","source":{"kind":"arxiv","id":"1206.2384","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.2384","created_at":"2026-05-18T03:29:27Z"},{"alias_kind":"arxiv_version","alias_value":"1206.2384v2","created_at":"2026-05-18T03:29:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.2384","created_at":"2026-05-18T03:29:27Z"},{"alias_kind":"pith_short_12","alias_value":"CRZY47XJ4HHV","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CRZY47XJ4HHVEB4T","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CRZY47XJ","created_at":"2026-05-18T12:27:01Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:CRZY47XJ4HHVEB4T6C56G44PWZ","target":"record","payload":{"canonical_record":{"source":{"id":"1206.2384","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-06-11T21:33:46Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"6e653b6660b14d4c5c76dac02be79ce9b5efb832612a3dd2de077abcb4a845f3","abstract_canon_sha256":"7dafad20b108133f126a061ae9ab054dffa58fcd1d65d0dc0622d4035ed186ab"},"schema_version":"1.0"},"canonical_sha256":"14738e7ee9e1cf520793f0bbe3738fb64dbe7bf614da88dce2e630cb85d82b86","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:27.476513Z","signature_b64":"iJRswj8r7cAkaP4lavBgH+cUUizayhJZgaiLpr7jXmWn7crd9qV0Fd6VacCz30lxYyVtOhgzcgyykzXdkf6bBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"14738e7ee9e1cf520793f0bbe3738fb64dbe7bf614da88dce2e630cb85d82b86","last_reissued_at":"2026-05-18T03:29:27.475915Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:27.475915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1206.2384","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":"2026-05-18T03:29:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"DhWxQGt7Ydf79GWgUEOGQHmd/sf5HjwuN/IheSZaRqo7RCmo7nsgg+kwqGauprtnvmTvfJ2omlDMDwQ2yWJvAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:46:57.993179Z"},"content_sha256":"51b08b4547fc42619a6cb5f8f167ba87a544efff346f04b0d3ec615b6dd272cf","schema_version":"1.0","event_id":"sha256:51b08b4547fc42619a6cb5f8f167ba87a544efff346f04b0d3ec615b6dd272cf"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:CRZY47XJ4HHVEB4T6C56G44PWZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Bounding the fractional chromatic number of $K_\\Delta$-free graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Andrew D. King, Katherine Edwards","submitted_at":"2012-06-11T21:33:46Z","abstract_excerpt":"King, Lu, and Peng recently proved that for $\\Delta\\geq 4$, any $K_\\Delta$-free graph with maximum degree $\\Delta$ has fractional chromatic number at most $\\Delta-\\tfrac{2}{67}$ unless it is isomorphic to $C_5\\boxtimes K_2$ or $C_8^2$. Using a different approach we give improved bounds for $\\Delta\\geq 6$ and pose several related conjectures. Our proof relies on a weighted local generalization of the fractional relaxation of Reed's $\\omega$, $\\Delta$, $\\chi$ conjecture."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2384","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":"2026-05-18T03:29:27Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WNxYi4Zkb0/Gw0gXdnL9m3JJzXRzK/KR5bMpIxIvn4baWdHzshRBB34BZ0yjoydpEThdg3lveKP15I6gWC5dAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T14:46:57.993873Z"},"content_sha256":"d39e2393ac45678bc97d43c54161e376dbc5af22f768a786be30c5ce2ea2a024","schema_version":"1.0","event_id":"sha256:d39e2393ac45678bc97d43c54161e376dbc5af22f768a786be30c5ce2ea2a024"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/CRZY47XJ4HHVEB4T6C56G44PWZ/bundle.json","state_url":"https://pith.science/pith/CRZY47XJ4HHVEB4T6C56G44PWZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/CRZY47XJ4HHVEB4T6C56G44PWZ/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":"2026-05-27T14:46:57Z","links":{"resolver":"https://pith.science/pith/CRZY47XJ4HHVEB4T6C56G44PWZ","bundle":"https://pith.science/pith/CRZY47XJ4HHVEB4T6C56G44PWZ/bundle.json","state":"https://pith.science/pith/CRZY47XJ4HHVEB4T6C56G44PWZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/CRZY47XJ4HHVEB4T6C56G44PWZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:CRZY47XJ4HHVEB4T6C56G44PWZ","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":"7dafad20b108133f126a061ae9ab054dffa58fcd1d65d0dc0622d4035ed186ab","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-06-11T21:33:46Z","title_canon_sha256":"6e653b6660b14d4c5c76dac02be79ce9b5efb832612a3dd2de077abcb4a845f3"},"schema_version":"1.0","source":{"id":"1206.2384","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1206.2384","created_at":"2026-05-18T03:29:27Z"},{"alias_kind":"arxiv_version","alias_value":"1206.2384v2","created_at":"2026-05-18T03:29:27Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1206.2384","created_at":"2026-05-18T03:29:27Z"},{"alias_kind":"pith_short_12","alias_value":"CRZY47XJ4HHV","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_16","alias_value":"CRZY47XJ4HHVEB4T","created_at":"2026-05-18T12:27:01Z"},{"alias_kind":"pith_short_8","alias_value":"CRZY47XJ","created_at":"2026-05-18T12:27:01Z"}],"graph_snapshots":[{"event_id":"sha256:d39e2393ac45678bc97d43c54161e376dbc5af22f768a786be30c5ce2ea2a024","target":"graph","created_at":"2026-05-18T03:29:27Z","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":"King, Lu, and Peng recently proved that for $\\Delta\\geq 4$, any $K_\\Delta$-free graph with maximum degree $\\Delta$ has fractional chromatic number at most $\\Delta-\\tfrac{2}{67}$ unless it is isomorphic to $C_5\\boxtimes K_2$ or $C_8^2$. Using a different approach we give improved bounds for $\\Delta\\geq 6$ and pose several related conjectures. Our proof relies on a weighted local generalization of the fractional relaxation of Reed's $\\omega$, $\\Delta$, $\\chi$ conjecture.","authors_text":"Andrew D. King, Katherine Edwards","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-06-11T21:33:46Z","title":"Bounding the fractional chromatic number of $K_\\Delta$-free graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1206.2384","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":"sha256:51b08b4547fc42619a6cb5f8f167ba87a544efff346f04b0d3ec615b6dd272cf","target":"record","created_at":"2026-05-18T03:29:27Z","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":"7dafad20b108133f126a061ae9ab054dffa58fcd1d65d0dc0622d4035ed186ab","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2012-06-11T21:33:46Z","title_canon_sha256":"6e653b6660b14d4c5c76dac02be79ce9b5efb832612a3dd2de077abcb4a845f3"},"schema_version":"1.0","source":{"id":"1206.2384","kind":"arxiv","version":2}},"canonical_sha256":"14738e7ee9e1cf520793f0bbe3738fb64dbe7bf614da88dce2e630cb85d82b86","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"14738e7ee9e1cf520793f0bbe3738fb64dbe7bf614da88dce2e630cb85d82b86","first_computed_at":"2026-05-18T03:29:27.475915Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:29:27.475915Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"iJRswj8r7cAkaP4lavBgH+cUUizayhJZgaiLpr7jXmWn7crd9qV0Fd6VacCz30lxYyVtOhgzcgyykzXdkf6bBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:29:27.476513Z","signed_message":"canonical_sha256_bytes"},"source_id":"1206.2384","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:51b08b4547fc42619a6cb5f8f167ba87a544efff346f04b0d3ec615b6dd272cf","sha256:d39e2393ac45678bc97d43c54161e376dbc5af22f768a786be30c5ce2ea2a024"],"state_sha256":"c8e984a86213f59852a8d867816e54d05c8e0572c0163a05253ddabc7452e25d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Yt3eXPaBmJbjvHiOyCnh5yjqGR1AceWhxyoia2JK94eV7RNFoxb32595z2OhtwSObcRGZmwvOa34R5TwofsbCg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T14:46:57.997036Z","bundle_sha256":"e5dae5dedd9ecc4a4fb5dccac26c2ea753cecd49134d9569097d252cd36a897d"}}