{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2015:GSNGNJPN7CT66T67BQQUGLC556","short_pith_number":"pith:GSNGNJPN","schema_version":"1.0","canonical_sha256":"349a66a5edf8a7ef4fdf0c21432c5def8deaa08a62c662afc2599bc56cfa4a18","source":{"kind":"arxiv","id":"1510.04463","version":1},"attestation_state":"computed","paper":{"title":"Betti numbers of subgraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Duc Ho, Huy Tai Ha","submitted_at":"2015-10-15T10:23:22Z","abstract_excerpt":"Let $G$ be a simple graph on $n$ vertices. Let $H$ be either the complete graph $K_m$ or the complete bipartite graph $K_{r,s}$ on a subset of the vertices in $G$. We show that $G$ contains $H$ as a subgraph if and only if $\\beta_{i,\\alpha}(H) \\le \\beta_{i,\\alpha}(G)$ for all $i \\ge 0$ and $\\alpha \\in \\mathbb{Z}^n$. In fact, it suffices to consider only the first syzygy module. In particular, we prove that $\\beta_{1,\\alpha}(H) \\le \\beta_{1,\\alpha}(G)$ for all $\\alpha \\in \\mathbb{Z}^n$ if and only if $G$ contains a subgraph that is isomorphic to either $H$ or a multipartite graph $K_{2,\\dots,2,"},"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":"1510.04463","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AC","submitted_at":"2015-10-15T10:23:22Z","cross_cats_sorted":[],"title_canon_sha256":"dbba9c6f3ca055ca7a994e234ddde4eaad7b7e4726608198ad1372d54db206df","abstract_canon_sha256":"d023848d046a64cb32a0b75dfea8fc69d4517ff94a72145cd7cc854849d3cc38"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:30:07.089734Z","signature_b64":"8kyqZdSIf1Bxko9Sh5HLzwtYtJ7eaThl+TS9W9Loqc4Me7AaSfzjr1oQArMyGm/fEIogJ2c6dxseNL25YZfDAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"349a66a5edf8a7ef4fdf0c21432c5def8deaa08a62c662afc2599bc56cfa4a18","last_reissued_at":"2026-05-18T01:30:07.089183Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:30:07.089183Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Betti numbers of subgraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AC","authors_text":"Duc Ho, Huy Tai Ha","submitted_at":"2015-10-15T10:23:22Z","abstract_excerpt":"Let $G$ be a simple graph on $n$ vertices. Let $H$ be either the complete graph $K_m$ or the complete bipartite graph $K_{r,s}$ on a subset of the vertices in $G$. We show that $G$ contains $H$ as a subgraph if and only if $\\beta_{i,\\alpha}(H) \\le \\beta_{i,\\alpha}(G)$ for all $i \\ge 0$ and $\\alpha \\in \\mathbb{Z}^n$. In fact, it suffices to consider only the first syzygy module. In particular, we prove that $\\beta_{1,\\alpha}(H) \\le \\beta_{1,\\alpha}(G)$ for all $\\alpha \\in \\mathbb{Z}^n$ if and only if $G$ contains a subgraph that is isomorphic to either $H$ or a multipartite graph $K_{2,\\dots,2,"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.04463","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":"1510.04463","created_at":"2026-05-18T01:30:07.089268+00:00"},{"alias_kind":"arxiv_version","alias_value":"1510.04463v1","created_at":"2026-05-18T01:30:07.089268+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.04463","created_at":"2026-05-18T01:30:07.089268+00:00"},{"alias_kind":"pith_short_12","alias_value":"GSNGNJPN7CT6","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_16","alias_value":"GSNGNJPN7CT66T67","created_at":"2026-05-18T12:29:22.688609+00:00"},{"alias_kind":"pith_short_8","alias_value":"GSNGNJPN","created_at":"2026-05-18T12:29:22.688609+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/GSNGNJPN7CT66T67BQQUGLC556","json":"https://pith.science/pith/GSNGNJPN7CT66T67BQQUGLC556.json","graph_json":"https://pith.science/api/pith-number/GSNGNJPN7CT66T67BQQUGLC556/graph.json","events_json":"https://pith.science/api/pith-number/GSNGNJPN7CT66T67BQQUGLC556/events.json","paper":"https://pith.science/paper/GSNGNJPN"},"agent_actions":{"view_html":"https://pith.science/pith/GSNGNJPN7CT66T67BQQUGLC556","download_json":"https://pith.science/pith/GSNGNJPN7CT66T67BQQUGLC556.json","view_paper":"https://pith.science/paper/GSNGNJPN","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1510.04463&json=true","fetch_graph":"https://pith.science/api/pith-number/GSNGNJPN7CT66T67BQQUGLC556/graph.json","fetch_events":"https://pith.science/api/pith-number/GSNGNJPN7CT66T67BQQUGLC556/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/GSNGNJPN7CT66T67BQQUGLC556/action/timestamp_anchor","attest_storage":"https://pith.science/pith/GSNGNJPN7CT66T67BQQUGLC556/action/storage_attestation","attest_author":"https://pith.science/pith/GSNGNJPN7CT66T67BQQUGLC556/action/author_attestation","sign_citation":"https://pith.science/pith/GSNGNJPN7CT66T67BQQUGLC556/action/citation_signature","submit_replication":"https://pith.science/pith/GSNGNJPN7CT66T67BQQUGLC556/action/replication_record"}},"created_at":"2026-05-18T01:30:07.089268+00:00","updated_at":"2026-05-18T01:30:07.089268+00:00"}