{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:MC3BUOEI5BDH2YFD3KXPS6MUWQ","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":"b4fc1b64cdaf30f59b958ad6c8fbb527e1b0eee1bb547791b2a63b123f5a2fe2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-14T11:20:24Z","title_canon_sha256":"59de4e570c5c9b6c838fb59b6bad389819ad849b36a944b8f5d6e835fce98468"},"schema_version":"1.0","source":{"id":"1806.05468","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1806.05468","created_at":"2026-05-17T23:50:07Z"},{"alias_kind":"arxiv_version","alias_value":"1806.05468v2","created_at":"2026-05-17T23:50:07Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1806.05468","created_at":"2026-05-17T23:50:07Z"},{"alias_kind":"pith_short_12","alias_value":"MC3BUOEI5BDH","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_16","alias_value":"MC3BUOEI5BDH2YFD","created_at":"2026-05-18T12:32:37Z"},{"alias_kind":"pith_short_8","alias_value":"MC3BUOEI","created_at":"2026-05-18T12:32:37Z"}],"graph_snapshots":[{"event_id":"sha256:7a690ee136c38140e4123fe63d9e1f49ae618a78fa98dd11bc12e481eba20f8c","target":"graph","created_at":"2026-05-17T23:50:07Z","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":"We investigate the genus $g(n,m)$ of the Erd\\H{o}s-R\\'enyi random graph $G(n,m)$, providing a thorough description of how this relates to the function $m=m(n)$, and finding that there is different behaviour depending on which `region' $m$ falls into.\n  Results already exist for $m \\le \\frac{n}{2} + O(n^{2/3})$ and $m = \\omega \\left( n^{1+\\frac{1}{j}} \\right)$ for $j \\in \\mathbb{N}$, and so we focus on the intermediate cases. We establish that $g(n,m) = (1+o(1)) \\frac{m}{2}$ whp (with high probability) when $n \\ll m = n^{1+o(1)}$, that $g(n,m) = (1+o(1)) \\mu (\\lambda) m$ whp for a given functio","authors_text":"Chris Dowden, Michael Krivelevich, Mihyun Kang","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-14T11:20:24Z","title":"The genus of the Erd\\H{o}s-R\\'enyi random graph and the fragile genus property"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1806.05468","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":"sha256:65e18989ef6171b30bc20a9aec32240ab31a51848e11634a7c120d65a1e92adc","target":"record","created_at":"2026-05-17T23:50:07Z","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":"b4fc1b64cdaf30f59b958ad6c8fbb527e1b0eee1bb547791b2a63b123f5a2fe2","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2018-06-14T11:20:24Z","title_canon_sha256":"59de4e570c5c9b6c838fb59b6bad389819ad849b36a944b8f5d6e835fce98468"},"schema_version":"1.0","source":{"id":"1806.05468","kind":"arxiv","version":2}},"canonical_sha256":"60b61a3888e8467d60a3daaef97994b42a43a0b18c21e983c9fead84d633f8dd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"60b61a3888e8467d60a3daaef97994b42a43a0b18c21e983c9fead84d633f8dd","first_computed_at":"2026-05-17T23:50:07.219854Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:50:07.219854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"TLZd2NZBh0saZetIW+jiJeLr6YfFgydr9q2NFt+1obX1XpR2IdTDB+InwNwkLNunTQz1fxu7eg2WEdBjTvB2BA==","signature_status":"signed_v1","signed_at":"2026-05-17T23:50:07.220502Z","signed_message":"canonical_sha256_bytes"},"source_id":"1806.05468","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:65e18989ef6171b30bc20a9aec32240ab31a51848e11634a7c120d65a1e92adc","sha256:7a690ee136c38140e4123fe63d9e1f49ae618a78fa98dd11bc12e481eba20f8c"],"state_sha256":"559cdb4613739835cba5991a8e748d763848894b2c542b2fc2ae677ac9cbc207"}