{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:26VH7LWB7O5ZZR2X2OSOBU237H","short_pith_number":"pith:26VH7LWB","canonical_record":{"source":{"id":"1712.03196","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-08T17:48:19Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"aedc27fdc83d5fbc15901971d5440ed6d00105e59f65ddf8f80ed104c4254403","abstract_canon_sha256":"57ec7b8b834d57d99db2c40dc6d801eaa49d49c5aa0749988b8b785921da860e"},"schema_version":"1.0"},"canonical_sha256":"d7aa7faec1fbbb9cc757d3a4e0d35bf9d8745e8cbf4b7c006e57dd83ff4dc059","source":{"kind":"arxiv","id":"1712.03196","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.03196","created_at":"2026-05-17T23:46:19Z"},{"alias_kind":"arxiv_version","alias_value":"1712.03196v2","created_at":"2026-05-17T23:46:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.03196","created_at":"2026-05-17T23:46:19Z"},{"alias_kind":"pith_short_12","alias_value":"26VH7LWB7O5Z","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"26VH7LWB7O5ZZR2X","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"26VH7LWB","created_at":"2026-05-18T12:30:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:26VH7LWB7O5ZZR2X2OSOBU237H","target":"record","payload":{"canonical_record":{"source":{"id":"1712.03196","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-08T17:48:19Z","cross_cats_sorted":["math.AT"],"title_canon_sha256":"aedc27fdc83d5fbc15901971d5440ed6d00105e59f65ddf8f80ed104c4254403","abstract_canon_sha256":"57ec7b8b834d57d99db2c40dc6d801eaa49d49c5aa0749988b8b785921da860e"},"schema_version":"1.0"},"canonical_sha256":"d7aa7faec1fbbb9cc757d3a4e0d35bf9d8745e8cbf4b7c006e57dd83ff4dc059","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:46:19.904740Z","signature_b64":"mcP8MmaBXJgUv+Or5R5EGsoZzSzpzY/fyCMrbeJaHHUwemUlXSeKJpV7D526VkTXucXjvp3bFf+bfVq6Xa8WBQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"d7aa7faec1fbbb9cc757d3a4e0d35bf9d8745e8cbf4b7c006e57dd83ff4dc059","last_reissued_at":"2026-05-17T23:46:19.904002Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:46:19.904002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1712.03196","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:46:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ju45fGUZKNy7NTT3GcEdRfA8B/qiA27CSHUMTOukXQ8RhSOANP9aALZZAQt+fFhXHTfK+UpDIR0/uAEY6JpvCQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T15:06:31.709167Z"},"content_sha256":"3e0c4780c168f08f1a0163e71fb9ee2ec77db534202f4a89ae64d6f7849c4207","schema_version":"1.0","event_id":"sha256:3e0c4780c168f08f1a0163e71fb9ee2ec77db534202f4a89ae64d6f7849c4207"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:26VH7LWB7O5ZZR2X2OSOBU237H","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On inverse powers of graphs and topological implications of Hedetniemi's conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AT"],"primary_cat":"math.CO","authors_text":"Marcin Wrochna","submitted_at":"2017-12-08T17:48:19Z","abstract_excerpt":"We consider a natural graph operation $\\Omega_k$ that is a certain inverse (formally: the right adjoint) to taking the k-th power of a graph. We show that it preserves the topology (the $\\mathbb{Z}_2$-homotopy type) of the box complex, a basic tool in topological combinatorics. Moreover, we prove that the box complex of a graph G admits a $\\mathbb{Z}_2$-map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph $\\Omega_k(G)$ admits a homomorphism to H, for high enough k.\n  This allows to show that if Hedetniemi's conjecture on the chromatic number of graph pr"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.03196","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-17T23:46:19Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IVHeqRs0oyplLpxlz6hK+WYyVIoMf4CGuZP1Ii7Y+KhipY3mcDoyVHXi3rOYlX2fTb/pkRFd8PTGQH9nxQqvDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-29T15:06:31.709833Z"},"content_sha256":"59e7af1e0d337a006352d1fe04f85e60b0e15d897526fab62cf411413d8f10f0","schema_version":"1.0","event_id":"sha256:59e7af1e0d337a006352d1fe04f85e60b0e15d897526fab62cf411413d8f10f0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/26VH7LWB7O5ZZR2X2OSOBU237H/bundle.json","state_url":"https://pith.science/pith/26VH7LWB7O5ZZR2X2OSOBU237H/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/26VH7LWB7O5ZZR2X2OSOBU237H/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":"2026-05-29T15:06:31Z","links":{"resolver":"https://pith.science/pith/26VH7LWB7O5ZZR2X2OSOBU237H","bundle":"https://pith.science/pith/26VH7LWB7O5ZZR2X2OSOBU237H/bundle.json","state":"https://pith.science/pith/26VH7LWB7O5ZZR2X2OSOBU237H/state.json","well_known_bundle":"https://pith.science/.well-known/pith/26VH7LWB7O5ZZR2X2OSOBU237H/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:26VH7LWB7O5ZZR2X2OSOBU237H","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":"57ec7b8b834d57d99db2c40dc6d801eaa49d49c5aa0749988b8b785921da860e","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-08T17:48:19Z","title_canon_sha256":"aedc27fdc83d5fbc15901971d5440ed6d00105e59f65ddf8f80ed104c4254403"},"schema_version":"1.0","source":{"id":"1712.03196","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1712.03196","created_at":"2026-05-17T23:46:19Z"},{"alias_kind":"arxiv_version","alias_value":"1712.03196v2","created_at":"2026-05-17T23:46:19Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.03196","created_at":"2026-05-17T23:46:19Z"},{"alias_kind":"pith_short_12","alias_value":"26VH7LWB7O5Z","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_16","alias_value":"26VH7LWB7O5ZZR2X","created_at":"2026-05-18T12:30:55Z"},{"alias_kind":"pith_short_8","alias_value":"26VH7LWB","created_at":"2026-05-18T12:30:55Z"}],"graph_snapshots":[{"event_id":"sha256:59e7af1e0d337a006352d1fe04f85e60b0e15d897526fab62cf411413d8f10f0","target":"graph","created_at":"2026-05-17T23:46:19Z","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 consider a natural graph operation $\\Omega_k$ that is a certain inverse (formally: the right adjoint) to taking the k-th power of a graph. We show that it preserves the topology (the $\\mathbb{Z}_2$-homotopy type) of the box complex, a basic tool in topological combinatorics. Moreover, we prove that the box complex of a graph G admits a $\\mathbb{Z}_2$-map (an equivariant, continuous map) to the box complex of a graph H if and only if the graph $\\Omega_k(G)$ admits a homomorphism to H, for high enough k.\n  This allows to show that if Hedetniemi's conjecture on the chromatic number of graph pr","authors_text":"Marcin Wrochna","cross_cats":["math.AT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-08T17:48:19Z","title":"On inverse powers of graphs and topological implications of Hedetniemi's conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.03196","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":"sha256:3e0c4780c168f08f1a0163e71fb9ee2ec77db534202f4a89ae64d6f7849c4207","target":"record","created_at":"2026-05-17T23:46:19Z","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":"57ec7b8b834d57d99db2c40dc6d801eaa49d49c5aa0749988b8b785921da860e","cross_cats_sorted":["math.AT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-08T17:48:19Z","title_canon_sha256":"aedc27fdc83d5fbc15901971d5440ed6d00105e59f65ddf8f80ed104c4254403"},"schema_version":"1.0","source":{"id":"1712.03196","kind":"arxiv","version":2}},"canonical_sha256":"d7aa7faec1fbbb9cc757d3a4e0d35bf9d8745e8cbf4b7c006e57dd83ff4dc059","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d7aa7faec1fbbb9cc757d3a4e0d35bf9d8745e8cbf4b7c006e57dd83ff4dc059","first_computed_at":"2026-05-17T23:46:19.904002Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-17T23:46:19.904002Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"mcP8MmaBXJgUv+Or5R5EGsoZzSzpzY/fyCMrbeJaHHUwemUlXSeKJpV7D526VkTXucXjvp3bFf+bfVq6Xa8WBQ==","signature_status":"signed_v1","signed_at":"2026-05-17T23:46:19.904740Z","signed_message":"canonical_sha256_bytes"},"source_id":"1712.03196","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:3e0c4780c168f08f1a0163e71fb9ee2ec77db534202f4a89ae64d6f7849c4207","sha256:59e7af1e0d337a006352d1fe04f85e60b0e15d897526fab62cf411413d8f10f0"],"state_sha256":"63f834b3ed8c848947d7af9eb186b8e9beabf5d0af5016798d6f2d58863d3de2"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"m54aoiskTljxAGwiJ4Vh5tl+TKC4whquX7psT6gown4cHQeugtVZEV9t+xz4mDgz1PP2S496VDaL9VFeAEi5Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-29T15:06:31.712053Z","bundle_sha256":"a89494f20175c8ec5e552986adcf5536b3454c2dc0e013eaf6e07b418e03e30d"}}