{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:A2O6EWWU2R4AORB5F7VODEDVV5","short_pith_number":"pith:A2O6EWWU","schema_version":"1.0","canonical_sha256":"069de25ad4d47807443d2feae19075af6cb8c4e389e31c15d18fbb21dbd419e7","source":{"kind":"arxiv","id":"1611.06031","version":1},"attestation_state":"computed","paper":{"title":"Equitable coloring of sparse planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D. Christopher Stephens, Gexin Yu, Jean-S\\'ebastien Sereni, Rong Luo","submitted_at":"2016-11-18T10:24:37Z","abstract_excerpt":"A proper vertex coloring of a graph $G$ is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold $\\chi_{eq}^*(G)$ of $G$ is the smallest integer $m$ such that $G$ is equitably $n$-colorable for all $n\\ge m$. We show that for planar graphs $G$ with minimum degree at least two, $\\chi_{eq}^*(G)\\le 4$ if the girth of $G$ is at least $10$, and $\\chi_{eq}^*(G)\\le 3$ if the girth of $G$ is at least $14$."},"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":"1611.06031","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-18T10:24:37Z","cross_cats_sorted":[],"title_canon_sha256":"2aac4752d64d520a66596725c88be9a202d73fd929e1b80395c90889d820ace1","abstract_canon_sha256":"ba8ce385d5412fd7d3de60ecb33e7783410c151344694119d7333d22fbb6e0e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:57:43.258443Z","signature_b64":"VQ1FDDl8xKVOEnZD2k2qaRthAd8DNA7r6TsmmnVfPVXrwStcdwFOJBxV/RpnWoUooyHgD84c2f9/FMHmafkEBw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"069de25ad4d47807443d2feae19075af6cb8c4e389e31c15d18fbb21dbd419e7","last_reissued_at":"2026-05-18T00:57:43.257794Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:57:43.257794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equitable coloring of sparse planar graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"D. Christopher Stephens, Gexin Yu, Jean-S\\'ebastien Sereni, Rong Luo","submitted_at":"2016-11-18T10:24:37Z","abstract_excerpt":"A proper vertex coloring of a graph $G$ is equitable if the sizes of color classes differ by at most one. The equitable chromatic threshold $\\chi_{eq}^*(G)$ of $G$ is the smallest integer $m$ such that $G$ is equitably $n$-colorable for all $n\\ge m$. We show that for planar graphs $G$ with minimum degree at least two, $\\chi_{eq}^*(G)\\le 4$ if the girth of $G$ is at least $10$, and $\\chi_{eq}^*(G)\\le 3$ if the girth of $G$ is at least $14$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06031","kind":"arxiv","version":1},"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":"1611.06031","created_at":"2026-05-18T00:57:43.257889+00:00"},{"alias_kind":"arxiv_version","alias_value":"1611.06031v1","created_at":"2026-05-18T00:57:43.257889+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.06031","created_at":"2026-05-18T00:57:43.257889+00:00"},{"alias_kind":"pith_short_12","alias_value":"A2O6EWWU2R4A","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_16","alias_value":"A2O6EWWU2R4AORB5","created_at":"2026-05-18T12:30:04.600751+00:00"},{"alias_kind":"pith_short_8","alias_value":"A2O6EWWU","created_at":"2026-05-18T12:30:04.600751+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/A2O6EWWU2R4AORB5F7VODEDVV5","json":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5.json","graph_json":"https://pith.science/api/pith-number/A2O6EWWU2R4AORB5F7VODEDVV5/graph.json","events_json":"https://pith.science/api/pith-number/A2O6EWWU2R4AORB5F7VODEDVV5/events.json","paper":"https://pith.science/paper/A2O6EWWU"},"agent_actions":{"view_html":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5","download_json":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5.json","view_paper":"https://pith.science/paper/A2O6EWWU","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1611.06031&json=true","fetch_graph":"https://pith.science/api/pith-number/A2O6EWWU2R4AORB5F7VODEDVV5/graph.json","fetch_events":"https://pith.science/api/pith-number/A2O6EWWU2R4AORB5F7VODEDVV5/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/action/timestamp_anchor","attest_storage":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/action/storage_attestation","attest_author":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/action/author_attestation","sign_citation":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/action/citation_signature","submit_replication":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/action/replication_record"}},"created_at":"2026-05-18T00:57:43.257889+00:00","updated_at":"2026-05-18T00:57:43.257889+00:00"}