{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:P3OZAP4S4MKC4L35CINFFXQVO4","short_pith_number":"pith:P3OZAP4S","canonical_record":{"source":{"id":"1601.07856","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-28T18:39:02Z","cross_cats_sorted":[],"title_canon_sha256":"4b7a743a87bf2a832aaae0bad5486d87716f75d9e2c432548841bd8ff284a950","abstract_canon_sha256":"d50baa1558808b891a40bceccf498c68a23e2ce375a617a83ca1efc5915451f0"},"schema_version":"1.0"},"canonical_sha256":"7edd903f92e3142e2f7d121a52de15771b7d296ae10a9e6f7fe2b55dec02f0e8","source":{"kind":"arxiv","id":"1601.07856","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.07856","created_at":"2026-05-18T01:20:06Z"},{"alias_kind":"arxiv_version","alias_value":"1601.07856v2","created_at":"2026-05-18T01:20:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07856","created_at":"2026-05-18T01:20:06Z"},{"alias_kind":"pith_short_12","alias_value":"P3OZAP4S4MKC","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"P3OZAP4S4MKC4L35","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"P3OZAP4S","created_at":"2026-05-18T12:30:36Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:P3OZAP4S4MKC4L35CINFFXQVO4","target":"record","payload":{"canonical_record":{"source":{"id":"1601.07856","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-28T18:39:02Z","cross_cats_sorted":[],"title_canon_sha256":"4b7a743a87bf2a832aaae0bad5486d87716f75d9e2c432548841bd8ff284a950","abstract_canon_sha256":"d50baa1558808b891a40bceccf498c68a23e2ce375a617a83ca1efc5915451f0"},"schema_version":"1.0"},"canonical_sha256":"7edd903f92e3142e2f7d121a52de15771b7d296ae10a9e6f7fe2b55dec02f0e8","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:20:06.235996Z","signature_b64":"tEVLLKPuyhPTcscRsiO7VP8o63aCS9Y/QNvp3le+dnQCGYR2KF0L8uigJMdk8n1UwCfSXAUSNTTuGo8BWMJDAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"7edd903f92e3142e2f7d121a52de15771b7d296ae10a9e6f7fe2b55dec02f0e8","last_reissued_at":"2026-05-18T01:20:06.235237Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:20:06.235237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.07856","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":"2026-05-18T01:20:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HqmT7ZifNj0vN89bvxKQF5yatsl6s5SX1OVMQVsTnp6ger5CeSPSV7LJUCAku6/dkNYnJ9pQjKK4jQzTbwd+Dg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:26:17.996908Z"},"content_sha256":"add5c0a643f2ce3cdea3a17ee8b66253a80846858a2004a52c0865f21572464c","schema_version":"1.0","event_id":"sha256:add5c0a643f2ce3cdea3a17ee8b66253a80846858a2004a52c0865f21572464c"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:P3OZAP4S4MKC4L35CINFFXQVO4","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Topologically $4$-chromatic graphs and signatures of odd cycles","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Claude Tardif, David Wehlau, Gord Simons","submitted_at":"2016-01-28T18:39:02Z","abstract_excerpt":"We investigate group-theoretic \"signatures\" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07856","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":"2026-05-18T01:20:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"BxeyA0FEuQSTiHE0qpZLQ9GO0Gm5R9ynEgpvxCMLPGPapkyEjAs4L6D+REssFLHBC481ozzrDhZw0rNOVksuCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T03:26:17.997257Z"},"content_sha256":"d6f176ee86e1f1caa4862d7a3d18cd6f3de5867aa72562ce0979d6b7a94ce233","schema_version":"1.0","event_id":"sha256:d6f176ee86e1f1caa4862d7a3d18cd6f3de5867aa72562ce0979d6b7a94ce233"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/P3OZAP4S4MKC4L35CINFFXQVO4/bundle.json","state_url":"https://pith.science/pith/P3OZAP4S4MKC4L35CINFFXQVO4/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/P3OZAP4S4MKC4L35CINFFXQVO4/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":"2026-06-23T03:26:17Z","links":{"resolver":"https://pith.science/pith/P3OZAP4S4MKC4L35CINFFXQVO4","bundle":"https://pith.science/pith/P3OZAP4S4MKC4L35CINFFXQVO4/bundle.json","state":"https://pith.science/pith/P3OZAP4S4MKC4L35CINFFXQVO4/state.json","well_known_bundle":"https://pith.science/.well-known/pith/P3OZAP4S4MKC4L35CINFFXQVO4/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:P3OZAP4S4MKC4L35CINFFXQVO4","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":"d50baa1558808b891a40bceccf498c68a23e2ce375a617a83ca1efc5915451f0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-28T18:39:02Z","title_canon_sha256":"4b7a743a87bf2a832aaae0bad5486d87716f75d9e2c432548841bd8ff284a950"},"schema_version":"1.0","source":{"id":"1601.07856","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.07856","created_at":"2026-05-18T01:20:06Z"},{"alias_kind":"arxiv_version","alias_value":"1601.07856v2","created_at":"2026-05-18T01:20:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.07856","created_at":"2026-05-18T01:20:06Z"},{"alias_kind":"pith_short_12","alias_value":"P3OZAP4S4MKC","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_16","alias_value":"P3OZAP4S4MKC4L35","created_at":"2026-05-18T12:30:36Z"},{"alias_kind":"pith_short_8","alias_value":"P3OZAP4S","created_at":"2026-05-18T12:30:36Z"}],"graph_snapshots":[{"event_id":"sha256:d6f176ee86e1f1caa4862d7a3d18cd6f3de5867aa72562ce0979d6b7a94ce233","target":"graph","created_at":"2026-05-18T01:20:06Z","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 group-theoretic \"signatures\" of odd cycles of a graph, and their connections to topological obstructions to 3-colourability. In the case of signatures derived from free groups, we prove that the existence of an odd cycle with trivial signature is equivalent to having the coindex of the hom-complex at least 2 (which implies that the chromatic number is at least 4). In the case of signatures derived from elementary abelian 2-groups we prove that the existence of an odd cycle with trivial signature is a sufficient condition for having the index of the hom-complex at least 2 (which ","authors_text":"Claude Tardif, David Wehlau, Gord Simons","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-28T18:39:02Z","title":"Topologically $4$-chromatic graphs and signatures of odd cycles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.07856","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:add5c0a643f2ce3cdea3a17ee8b66253a80846858a2004a52c0865f21572464c","target":"record","created_at":"2026-05-18T01:20:06Z","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":"d50baa1558808b891a40bceccf498c68a23e2ce375a617a83ca1efc5915451f0","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-01-28T18:39:02Z","title_canon_sha256":"4b7a743a87bf2a832aaae0bad5486d87716f75d9e2c432548841bd8ff284a950"},"schema_version":"1.0","source":{"id":"1601.07856","kind":"arxiv","version":2}},"canonical_sha256":"7edd903f92e3142e2f7d121a52de15771b7d296ae10a9e6f7fe2b55dec02f0e8","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"7edd903f92e3142e2f7d121a52de15771b7d296ae10a9e6f7fe2b55dec02f0e8","first_computed_at":"2026-05-18T01:20:06.235237Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:20:06.235237Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"tEVLLKPuyhPTcscRsiO7VP8o63aCS9Y/QNvp3le+dnQCGYR2KF0L8uigJMdk8n1UwCfSXAUSNTTuGo8BWMJDAg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:20:06.235996Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.07856","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:add5c0a643f2ce3cdea3a17ee8b66253a80846858a2004a52c0865f21572464c","sha256:d6f176ee86e1f1caa4862d7a3d18cd6f3de5867aa72562ce0979d6b7a94ce233"],"state_sha256":"7b1ba964e84a359098f74da93e545295b709f6949378bfc1f05b82a57b2b6cfc"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WnTZuUegM8YvXFvq1HnuM6cGWIkVEdReK+GW/pQ/TEPtyt+Q4ke4G/jJqR5RsgHE2RJXPjA8ujh1QjhZqxbIBg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T03:26:17.999534Z","bundle_sha256":"0253274b701a43fce890f0e622f2ecc9d46cd16960919cb8cee6722aed500082"}}