{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:A2O6EWWU2R4AORB5F7VODEDVV5","short_pith_number":"pith:A2O6EWWU","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"},"canonical_sha256":"069de25ad4d47807443d2feae19075af6cb8c4e389e31c15d18fbb21dbd419e7","source":{"kind":"arxiv","id":"1611.06031","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.06031","created_at":"2026-05-18T00:57:43Z"},{"alias_kind":"arxiv_version","alias_value":"1611.06031v1","created_at":"2026-05-18T00:57:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.06031","created_at":"2026-05-18T00:57:43Z"},{"alias_kind":"pith_short_12","alias_value":"A2O6EWWU2R4A","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"A2O6EWWU2R4AORB5","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"A2O6EWWU","created_at":"2026-05-18T12:30:04Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:A2O6EWWU2R4AORB5F7VODEDVV5","target":"record","payload":{"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"},"canonical_sha256":"069de25ad4d47807443d2feae19075af6cb8c4e389e31c15d18fbb21dbd419e7","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"},"source_kind":"arxiv","source_id":"1611.06031","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:57:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8Uzo24VzaNJk12gk6vvfYFUtK+5rB2YqPX91wZhWO43rQpMvpgMwe34p+lP+o/gf1DjSZccCO0Z4hH/MoAzhAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:15:31.571432Z"},"content_sha256":"81340bbc28e0e18fcb7bb459af63a41f9f9f04e4063e22f6a58f9144bd6441a0","schema_version":"1.0","event_id":"sha256:81340bbc28e0e18fcb7bb459af63a41f9f9f04e4063e22f6a58f9144bd6441a0"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:A2O6EWWU2R4AORB5F7VODEDVV5","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:57:43Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Smd+O2H8d8Bqxsl7zXMsQ0lk0K9LSttMX1z/gGM57pUCypI6YYeClqPoOgV705rKUgaUvdbtBd6UJjMmpUK0Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T08:15:31.571803Z"},"content_sha256":"3a615fa10f6cd9f7cc406fb6678a589066598d182805e4437499e9e06b8cc568","schema_version":"1.0","event_id":"sha256:3a615fa10f6cd9f7cc406fb6678a589066598d182805e4437499e9e06b8cc568"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/bundle.json","state_url":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/A2O6EWWU2R4AORB5F7VODEDVV5/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":"2026-06-02T08:15:31Z","links":{"resolver":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5","bundle":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/bundle.json","state":"https://pith.science/pith/A2O6EWWU2R4AORB5F7VODEDVV5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/A2O6EWWU2R4AORB5F7VODEDVV5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:A2O6EWWU2R4AORB5F7VODEDVV5","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":"ba8ce385d5412fd7d3de60ecb33e7783410c151344694119d7333d22fbb6e0e9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-18T10:24:37Z","title_canon_sha256":"2aac4752d64d520a66596725c88be9a202d73fd929e1b80395c90889d820ace1"},"schema_version":"1.0","source":{"id":"1611.06031","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1611.06031","created_at":"2026-05-18T00:57:43Z"},{"alias_kind":"arxiv_version","alias_value":"1611.06031v1","created_at":"2026-05-18T00:57:43Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1611.06031","created_at":"2026-05-18T00:57:43Z"},{"alias_kind":"pith_short_12","alias_value":"A2O6EWWU2R4A","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"A2O6EWWU2R4AORB5","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"A2O6EWWU","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:3a615fa10f6cd9f7cc406fb6678a589066598d182805e4437499e9e06b8cc568","target":"graph","created_at":"2026-05-18T00:57:43Z","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":"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 $14$.","authors_text":"D. Christopher Stephens, Gexin Yu, Jean-S\\'ebastien Sereni, Rong Luo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-18T10:24:37Z","title":"Equitable coloring of sparse planar graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.06031","kind":"arxiv","version":1},"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":"sha256:81340bbc28e0e18fcb7bb459af63a41f9f9f04e4063e22f6a58f9144bd6441a0","target":"record","created_at":"2026-05-18T00:57:43Z","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":"ba8ce385d5412fd7d3de60ecb33e7783410c151344694119d7333d22fbb6e0e9","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-11-18T10:24:37Z","title_canon_sha256":"2aac4752d64d520a66596725c88be9a202d73fd929e1b80395c90889d820ace1"},"schema_version":"1.0","source":{"id":"1611.06031","kind":"arxiv","version":1}},"canonical_sha256":"069de25ad4d47807443d2feae19075af6cb8c4e389e31c15d18fbb21dbd419e7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"069de25ad4d47807443d2feae19075af6cb8c4e389e31c15d18fbb21dbd419e7","first_computed_at":"2026-05-18T00:57:43.257794Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:57:43.257794Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"VQ1FDDl8xKVOEnZD2k2qaRthAd8DNA7r6TsmmnVfPVXrwStcdwFOJBxV/RpnWoUooyHgD84c2f9/FMHmafkEBw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:57:43.258443Z","signed_message":"canonical_sha256_bytes"},"source_id":"1611.06031","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:81340bbc28e0e18fcb7bb459af63a41f9f9f04e4063e22f6a58f9144bd6441a0","sha256:3a615fa10f6cd9f7cc406fb6678a589066598d182805e4437499e9e06b8cc568"],"state_sha256":"5f6bfd34b19c82517428f82435b4125c3bd2a7ef0ea153d8b22c44e9e6fe1a71"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"SsYzOhA6NwesfyzwBk+kjC3S+3vh3bwcrTMzeXxjmaSk5zglPsNRKGWPDrCP08o+jMLE6SnVw6t5VdtiB8+qDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T08:15:31.573754Z","bundle_sha256":"1ab8d04e87fe4bc3b0af32413c94fe7934a69f4588b25185528814c8750ff0a8"}}