{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:WAFGMMDIFTCMJXHOGSE22X64DG","short_pith_number":"pith:WAFGMMDI","schema_version":"1.0","canonical_sha256":"b00a6630682cc4c4dcee3489ad5fdc199a2110e617c16e1f3cc1894daeea30c5","source":{"kind":"arxiv","id":"1601.07854","version":2},"attestation_state":"computed","paper":{"title":"Homomorphism Complexes and k-Cores","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Greg Malen","submitted_at":"2016-01-28T18:28:06Z","abstract_excerpt":"We prove that the topological connectivity of a graph homomorphism complex Hom($G,K_m$) is at least $m-D(G)-2$, where $\\displaystyle D(G)=\\max_{H\\subseteq G}\\delta(H)$. This is a strong generalization of a theorem of Cuki\\'{c} and Kozlov, in which $D(G)$ is replaced by the maximum degree $\\Delta(G)$. It also generalizes the graph theoretic bound for chromatic number, $\\displaystyle\\chi(G)\\leq D(G)+1$, as $\\displaystyle\\chi(G)=\\min\\{ m:\\text{Hom}(G,K_m)\\neq\\varnothing\\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1601.07854","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-28T18:28:06Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"73ee8f60e3801ad13c0dae63b4a5fd4447a1bfb1f7eaef009f075837da47e308","abstract_canon_sha256":"0bd0e721a3aec307831293985e8ba5b62263b0a9493639f902c0d3a4a2714924"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:51.664795Z","signature_b64":"7JfpyUH2sc1w6gsdlnUV+7AsLeZaQq3Z5WQUE2g7/3oxsJnkTZDQLCGEtD5RyVhMV8gt0mmNbMnncF4U3sFfBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b00a6630682cc4c4dcee3489ad5fdc199a2110e617c16e1f3cc1894daeea30c5","last_reissued_at":"2026-05-18T01:20:51.664047Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:51.664047Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Homomorphism Complexes and k-Cores","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Greg Malen","submitted_at":"2016-01-28T18:28:06Z","abstract_excerpt":"We prove that the topological connectivity of a graph homomorphism complex Hom($G,K_m$) is at least $m-D(G)-2$, where $\\displaystyle D(G)=\\max_{H\\subseteq G}\\delta(H)$. This is a strong generalization of a theorem of Cuki\\'{c} and Kozlov, in which $D(G)$ is replaced by the maximum degree $\\Delta(G)$. It also generalizes the graph theoretic bound for chromatic number, $\\displaystyle\\chi(G)\\leq D(G)+1$, as $\\displaystyle\\chi(G)=\\min\\{ m:\\text{Hom}(G,K_m)\\neq\\varnothing\\}$. Furthermore, we use this result to examine homological phase transitions in the random polyhedral complexes Hom$(G(n,p),K_m)"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07854","kind":"arxiv","version":2},"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":"1601.07854","created_at":"2026-05-18T01:20:51.664173+00:00"},{"alias_kind":"arxiv_version","alias_value":"1601.07854v2","created_at":"2026-05-18T01:20:51.664173+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07854","created_at":"2026-05-18T01:20:51.664173+00:00"},{"alias_kind":"pith_short_12","alias_value":"WAFGMMDIFTCM","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_16","alias_value":"WAFGMMDIFTCMJXHO","created_at":"2026-05-18T12:30:48.956258+00:00"},{"alias_kind":"pith_short_8","alias_value":"WAFGMMDI","created_at":"2026-05-18T12:30:48.956258+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/WAFGMMDIFTCMJXHOGSE22X64DG","json":"https://pith.science/pith/WAFGMMDIFTCMJXHOGSE22X64DG.json","graph_json":"https://pith.science/api/pith-number/WAFGMMDIFTCMJXHOGSE22X64DG/graph.json","events_json":"https://pith.science/api/pith-number/WAFGMMDIFTCMJXHOGSE22X64DG/events.json","paper":"https://pith.science/paper/WAFGMMDI"},"agent_actions":{"view_html":"https://pith.science/pith/WAFGMMDIFTCMJXHOGSE22X64DG","download_json":"https://pith.science/pith/WAFGMMDIFTCMJXHOGSE22X64DG.json","view_paper":"https://pith.science/paper/WAFGMMDI","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1601.07854&json=true","fetch_graph":"https://pith.science/api/pith-number/WAFGMMDIFTCMJXHOGSE22X64DG/graph.json","fetch_events":"https://pith.science/api/pith-number/WAFGMMDIFTCMJXHOGSE22X64DG/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/WAFGMMDIFTCMJXHOGSE22X64DG/action/timestamp_anchor","attest_storage":"https://pith.science/pith/WAFGMMDIFTCMJXHOGSE22X64DG/action/storage_attestation","attest_author":"https://pith.science/pith/WAFGMMDIFTCMJXHOGSE22X64DG/action/author_attestation","sign_citation":"https://pith.science/pith/WAFGMMDIFTCMJXHOGSE22X64DG/action/citation_signature","submit_replication":"https://pith.science/pith/WAFGMMDIFTCMJXHOGSE22X64DG/action/replication_record"}},"created_at":"2026-05-18T01:20:51.664173+00:00","updated_at":"2026-05-18T01:20:51.664173+00:00"}