{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:4BA7DC6ECO6MDH4F3LM5M5TW7L","short_pith_number":"pith:4BA7DC6E","schema_version":"1.0","canonical_sha256":"e041f18bc413bcc19f85dad9d67676faf6536668da08a6c160cff5eff359869c","source":{"kind":"arxiv","id":"2606.00375","version":1},"attestation_state":"computed","paper":{"title":"Embeddings of critical graphs near the Heawood bound","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Timothy Sun","submitted_at":"2026-05-29T21:33:39Z","abstract_excerpt":"Complementing a theorem of \\v{S}krekovski, we characterize the $(h-1)$-critical graphs embeddable in surfaces of Euler genus at least $5$, where $h$ denotes the Heawood number of the surface. Outside of a few small cases, the bulk of our proof is determining the genus of the join of a complete graph and the 5-cycle. As a byproduct of our proof, we also provide a simpler solution to the minimum triangulations problem for nonorientable surfaces using the theory of current graphs."},"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":"2606.00375","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-05-29T21:33:39Z","cross_cats_sorted":[],"title_canon_sha256":"cf100331e9b8588afd7be7e5d02104c43ab4a2165c9ed036b16bb77cc5e4804a","abstract_canon_sha256":"6f1300ef80f7c80ecabe7540695a1b63d2c69c5593b948285da40883feb7121d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-02T01:03:52.956942Z","signature_b64":"MB4MWN+2dpyX7Bz44GJ3G0h+ZYBPR+VuZhfnCzhovF7rgufCer8vYk4FML9ZwZmNWTtYPy/UHjjUtsUswvntBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"e041f18bc413bcc19f85dad9d67676faf6536668da08a6c160cff5eff359869c","last_reissued_at":"2026-06-02T01:03:52.956512Z","signature_status":"signed_v1","first_computed_at":"2026-06-02T01:03:52.956512Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Embeddings of critical graphs near the Heawood bound","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Timothy Sun","submitted_at":"2026-05-29T21:33:39Z","abstract_excerpt":"Complementing a theorem of \\v{S}krekovski, we characterize the $(h-1)$-critical graphs embeddable in surfaces of Euler genus at least $5$, where $h$ denotes the Heawood number of the surface. Outside of a few small cases, the bulk of our proof is determining the genus of the join of a complete graph and the 5-cycle. As a byproduct of our proof, we also provide a simpler solution to the minimum triangulations problem for nonorientable surfaces using the theory of current graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.00375","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.00375/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2606.00375","created_at":"2026-06-02T01:03:52.956579+00:00"},{"alias_kind":"arxiv_version","alias_value":"2606.00375v1","created_at":"2026-06-02T01:03:52.956579+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.00375","created_at":"2026-06-02T01:03:52.956579+00:00"},{"alias_kind":"pith_short_12","alias_value":"4BA7DC6ECO6M","created_at":"2026-06-02T01:03:52.956579+00:00"},{"alias_kind":"pith_short_16","alias_value":"4BA7DC6ECO6MDH4F","created_at":"2026-06-02T01:03:52.956579+00:00"},{"alias_kind":"pith_short_8","alias_value":"4BA7DC6E","created_at":"2026-06-02T01:03:52.956579+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/4BA7DC6ECO6MDH4F3LM5M5TW7L","json":"https://pith.science/pith/4BA7DC6ECO6MDH4F3LM5M5TW7L.json","graph_json":"https://pith.science/api/pith-number/4BA7DC6ECO6MDH4F3LM5M5TW7L/graph.json","events_json":"https://pith.science/api/pith-number/4BA7DC6ECO6MDH4F3LM5M5TW7L/events.json","paper":"https://pith.science/paper/4BA7DC6E"},"agent_actions":{"view_html":"https://pith.science/pith/4BA7DC6ECO6MDH4F3LM5M5TW7L","download_json":"https://pith.science/pith/4BA7DC6ECO6MDH4F3LM5M5TW7L.json","view_paper":"https://pith.science/paper/4BA7DC6E","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2606.00375&json=true","fetch_graph":"https://pith.science/api/pith-number/4BA7DC6ECO6MDH4F3LM5M5TW7L/graph.json","fetch_events":"https://pith.science/api/pith-number/4BA7DC6ECO6MDH4F3LM5M5TW7L/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/4BA7DC6ECO6MDH4F3LM5M5TW7L/action/timestamp_anchor","attest_storage":"https://pith.science/pith/4BA7DC6ECO6MDH4F3LM5M5TW7L/action/storage_attestation","attest_author":"https://pith.science/pith/4BA7DC6ECO6MDH4F3LM5M5TW7L/action/author_attestation","sign_citation":"https://pith.science/pith/4BA7DC6ECO6MDH4F3LM5M5TW7L/action/citation_signature","submit_replication":"https://pith.science/pith/4BA7DC6ECO6MDH4F3LM5M5TW7L/action/replication_record"}},"created_at":"2026-06-02T01:03:52.956579+00:00","updated_at":"2026-06-02T01:03:52.956579+00:00"}