{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:RNXP24HFB7OLIIGG7PRSBKHWTO","short_pith_number":"pith:RNXP24HF","canonical_record":{"source":{"id":"1209.4510","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-20T12:28:12Z","cross_cats_sorted":[],"title_canon_sha256":"8e0dbd9d7cc38cdde2c3f673a51b5fb554a88606bf39ea75df7a01937d8349d3","abstract_canon_sha256":"1fc231411fdf67a1d674149e4caadd6b81c87fc8c3541516a87371ac1725ab8d"},"schema_version":"1.0"},"canonical_sha256":"8b6efd70e50fdcb420c6fbe320a8f69bb1d929031d4f51c90c3d8700761db102","source":{"kind":"arxiv","id":"1209.4510","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4510","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4510v3","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4510","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"pith_short_12","alias_value":"RNXP24HFB7OL","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RNXP24HFB7OLIIGG","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RNXP24HF","created_at":"2026-05-18T12:27:20Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:RNXP24HFB7OLIIGG7PRSBKHWTO","target":"record","payload":{"canonical_record":{"source":{"id":"1209.4510","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-20T12:28:12Z","cross_cats_sorted":[],"title_canon_sha256":"8e0dbd9d7cc38cdde2c3f673a51b5fb554a88606bf39ea75df7a01937d8349d3","abstract_canon_sha256":"1fc231411fdf67a1d674149e4caadd6b81c87fc8c3541516a87371ac1725ab8d"},"schema_version":"1.0"},"canonical_sha256":"8b6efd70e50fdcb420c6fbe320a8f69bb1d929031d4f51c90c3d8700761db102","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:32:42.124645Z","signature_b64":"dLzWvYBRWpHtbIo3yCLpABmzA6sNVQBV95Ba10jDew4PnYaK+9JjsNo5DaYGkLxp18Df2AeYfyG8m4cRZpNCAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8b6efd70e50fdcb420c6fbe320a8f69bb1d929031d4f51c90c3d8700761db102","last_reissued_at":"2026-05-18T01:32:42.124152Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:32:42.124152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1209.4510","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:32:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"XuHrVaLYlDcXlIRVCc7jfdygmxfaTFeyxlr47Y2BjpGcRg6P8LQQYjmZHfY8t7WBCYR3dXBCh7H9cXMEKIfaDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:44:13.387138Z"},"content_sha256":"b191076463dca0f1dd51b3b34140d3930f9081ee50c15a2f19ce9cd828233196","schema_version":"1.0","event_id":"sha256:b191076463dca0f1dd51b3b34140d3930f9081ee50c15a2f19ce9cd828233196"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:RNXP24HFB7OLIIGG7PRSBKHWTO","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"1-factor and cycle covers of cubic graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Eckhard Steffen","submitted_at":"2012-09-20T12:28:12Z","abstract_excerpt":"Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of $G$.\n  Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if $\\mu_3(G) \\not = 0$, then $2 \\mu_3(G)$ is an upper bound for the girth of $G$. We also prove some new upper bounds for the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4510","kind":"arxiv","version":3},"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":"2026-05-18T01:32:42Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tlIzS5ssaK0iNW3NwlSIDzDH8uH8vG1iGX83XIO6o+Nvo/uMegnQ9Vz6Mctt+tEVMlqY2KmMru1RVvjcHitVDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-22T12:44:13.387507Z"},"content_sha256":"6cb879c1bccfc3ebadbad422d40d0f5225a1931c17a8a2ce5817a6a246b4bece","schema_version":"1.0","event_id":"sha256:6cb879c1bccfc3ebadbad422d40d0f5225a1931c17a8a2ce5817a6a246b4bece"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/RNXP24HFB7OLIIGG7PRSBKHWTO/bundle.json","state_url":"https://pith.science/pith/RNXP24HFB7OLIIGG7PRSBKHWTO/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/RNXP24HFB7OLIIGG7PRSBKHWTO/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":"2026-06-22T12:44:13Z","links":{"resolver":"https://pith.science/pith/RNXP24HFB7OLIIGG7PRSBKHWTO","bundle":"https://pith.science/pith/RNXP24HFB7OLIIGG7PRSBKHWTO/bundle.json","state":"https://pith.science/pith/RNXP24HFB7OLIIGG7PRSBKHWTO/state.json","well_known_bundle":"https://pith.science/.well-known/pith/RNXP24HFB7OLIIGG7PRSBKHWTO/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:RNXP24HFB7OLIIGG7PRSBKHWTO","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":"1fc231411fdf67a1d674149e4caadd6b81c87fc8c3541516a87371ac1725ab8d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-20T12:28:12Z","title_canon_sha256":"8e0dbd9d7cc38cdde2c3f673a51b5fb554a88606bf39ea75df7a01937d8349d3"},"schema_version":"1.0","source":{"id":"1209.4510","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1209.4510","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"arxiv_version","alias_value":"1209.4510v3","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1209.4510","created_at":"2026-05-18T01:32:42Z"},{"alias_kind":"pith_short_12","alias_value":"RNXP24HFB7OL","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_16","alias_value":"RNXP24HFB7OLIIGG","created_at":"2026-05-18T12:27:20Z"},{"alias_kind":"pith_short_8","alias_value":"RNXP24HF","created_at":"2026-05-18T12:27:20Z"}],"graph_snapshots":[{"event_id":"sha256:6cb879c1bccfc3ebadbad422d40d0f5225a1931c17a8a2ce5817a6a246b4bece","target":"graph","created_at":"2026-05-18T01:32:42Z","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":"Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of $G$.\n  Any list of three 1-factors induces a core of a cubic graph. We use results on the structure of cores to prove sufficient conditions for Berge-covers and for the existence of three 1-factors with empty intersection. Furthermore, if $\\mu_3(G) \\not = 0$, then $2 \\mu_3(G)$ is an upper bound for the girth of $G$. We also prove some new upper bounds for the","authors_text":"Eckhard Steffen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-20T12:28:12Z","title":"1-factor and cycle covers of cubic graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1209.4510","kind":"arxiv","version":3},"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:b191076463dca0f1dd51b3b34140d3930f9081ee50c15a2f19ce9cd828233196","target":"record","created_at":"2026-05-18T01:32:42Z","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":"1fc231411fdf67a1d674149e4caadd6b81c87fc8c3541516a87371ac1725ab8d","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-09-20T12:28:12Z","title_canon_sha256":"8e0dbd9d7cc38cdde2c3f673a51b5fb554a88606bf39ea75df7a01937d8349d3"},"schema_version":"1.0","source":{"id":"1209.4510","kind":"arxiv","version":3}},"canonical_sha256":"8b6efd70e50fdcb420c6fbe320a8f69bb1d929031d4f51c90c3d8700761db102","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"8b6efd70e50fdcb420c6fbe320a8f69bb1d929031d4f51c90c3d8700761db102","first_computed_at":"2026-05-18T01:32:42.124152Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:32:42.124152Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dLzWvYBRWpHtbIo3yCLpABmzA6sNVQBV95Ba10jDew4PnYaK+9JjsNo5DaYGkLxp18Df2AeYfyG8m4cRZpNCAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:32:42.124645Z","signed_message":"canonical_sha256_bytes"},"source_id":"1209.4510","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b191076463dca0f1dd51b3b34140d3930f9081ee50c15a2f19ce9cd828233196","sha256:6cb879c1bccfc3ebadbad422d40d0f5225a1931c17a8a2ce5817a6a246b4bece"],"state_sha256":"a8862b01c9e5c1c5a517bb0208dca3a503ee48c36581cca429e5e17beb03f46f"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"dyaP6TPOHxiR4nB4K/Zer1OIMXJkzQrbEfaUsU1xWa/ESMkCDTSM+hZDfyrsRCnFGSSbWVa5bZuOZSx7SGfrAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-22T12:44:13.389482Z","bundle_sha256":"9c37d94bf3e1d78e1f4f057b0b289dd6b970e9d6ed8313117cabe8e95736797b"}}