{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2012:5KJ7DKV2GWAAEUFFJYPDSA654V","short_pith_number":"pith:5KJ7DKV2","schema_version":"1.0","canonical_sha256":"ea93f1aaba35800250a54e1e3903dde57db36afb5fb6d72ec1db919448d20413","source":{"kind":"arxiv","id":"1205.5405","version":2},"attestation_state":"computed","paper":{"title":"Extensions of Fractional Precolorings show Discontinuous Behavior","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Kral, Jan van den Heuvel, Jan Volec, Jean-Sebastien Sereni, Martin Kupec","submitted_at":"2012-05-24T11:42:45Z","abstract_excerpt":"We study the following problem: given a real number k and integer d, what is the smallest epsilon such that any fractional (k+epsilon)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+epsilon)-coloring of the whole graph? The exact values of epsilon were known for k=2 and k\\ge3 and any d. We determine the exact values of epsilon for k \\in (2,3) if d=4, and k \\in [2.5,3) if d=6, and give upper bounds for k \\in (2,3) if d=5,7, and k \\in (2,2.5) if d=6. Surprisingly, epsilon viewed as a function of k is discontinuous f"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1205.5405","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-24T11:42:45Z","cross_cats_sorted":[],"title_canon_sha256":"ee59483050c9eeaecf968d968498dc5caf50f4e90bb530f730299c10f40ea8c4","abstract_canon_sha256":"9ef6b382c8a4f8bf9c20f6662c86763002dbabbe585bc77a0bba32b608c2220a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:04:24.358769Z","signature_b64":"kZC3feGxE1B7RTN34jt/GtpCZEcxN2yCFftt9DmZGB1jBBjWW+tm7x94zcqZajWWdruSz6jsL2rxm25PR9BbAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ea93f1aaba35800250a54e1e3903dde57db36afb5fb6d72ec1db919448d20413","last_reissued_at":"2026-05-18T03:04:24.358292Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:04:24.358292Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Extensions of Fractional Precolorings show Discontinuous Behavior","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Daniel Kral, Jan van den Heuvel, Jan Volec, Jean-Sebastien Sereni, Martin Kupec","submitted_at":"2012-05-24T11:42:45Z","abstract_excerpt":"We study the following problem: given a real number k and integer d, what is the smallest epsilon such that any fractional (k+epsilon)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+epsilon)-coloring of the whole graph? The exact values of epsilon were known for k=2 and k\\ge3 and any d. We determine the exact values of epsilon for k \\in (2,3) if d=4, and k \\in [2.5,3) if d=6, and give upper bounds for k \\in (2,3) if d=5,7, and k \\in (2,2.5) if d=6. Surprisingly, epsilon viewed as a function of k is discontinuous f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5405","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1205.5405","created_at":"2026-05-18T03:04:24.358378+00:00"},{"alias_kind":"arxiv_version","alias_value":"1205.5405v2","created_at":"2026-05-18T03:04:24.358378+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5405","created_at":"2026-05-18T03:04:24.358378+00:00"},{"alias_kind":"pith_short_12","alias_value":"5KJ7DKV2GWAA","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_16","alias_value":"5KJ7DKV2GWAAEUFF","created_at":"2026-05-18T12:26:53.410803+00:00"},{"alias_kind":"pith_short_8","alias_value":"5KJ7DKV2","created_at":"2026-05-18T12:26:53.410803+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V","json":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V.json","graph_json":"https://pith.science/api/pith-number/5KJ7DKV2GWAAEUFFJYPDSA654V/graph.json","events_json":"https://pith.science/api/pith-number/5KJ7DKV2GWAAEUFFJYPDSA654V/events.json","paper":"https://pith.science/paper/5KJ7DKV2"},"agent_actions":{"view_html":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V","download_json":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V.json","view_paper":"https://pith.science/paper/5KJ7DKV2","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1205.5405&json=true","fetch_graph":"https://pith.science/api/pith-number/5KJ7DKV2GWAAEUFFJYPDSA654V/graph.json","fetch_events":"https://pith.science/api/pith-number/5KJ7DKV2GWAAEUFFJYPDSA654V/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V/action/storage_attestation","attest_author":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V/action/author_attestation","sign_citation":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V/action/citation_signature","submit_replication":"https://pith.science/pith/5KJ7DKV2GWAAEUFFJYPDSA654V/action/replication_record"}},"created_at":"2026-05-18T03:04:24.358378+00:00","updated_at":"2026-05-18T03:04:24.358378+00:00"}