{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:VJB7OOAYYOK2DU7UPN4J3VPIKA","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":"160561428bafcc85a9f91d16c1fa263e87d14a7607153c79858b07bb0360fbf5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-03-29T16:16:35Z","title_canon_sha256":"69d2cb332548977e2b1f60a294285c8f2830453c100f333204df40844947eed5"},"schema_version":"1.0","source":{"id":"1603.08827","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.08827","created_at":"2026-05-18T00:49:34Z"},{"alias_kind":"arxiv_version","alias_value":"1603.08827v1","created_at":"2026-05-18T00:49:34Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.08827","created_at":"2026-05-18T00:49:34Z"},{"alias_kind":"pith_short_12","alias_value":"VJB7OOAYYOK2","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_16","alias_value":"VJB7OOAYYOK2DU7U","created_at":"2026-05-18T12:30:48Z"},{"alias_kind":"pith_short_8","alias_value":"VJB7OOAY","created_at":"2026-05-18T12:30:48Z"}],"graph_snapshots":[{"event_id":"sha256:4f7b071cd20807113726bfdde3b9cb7dcb68af8a94d1bfb9a77ab254403c3203","target":"graph","created_at":"2026-05-18T00:49:34Z","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":"An $n$-bit Gray code is a sequence of all $n$-bit strings such that consecutive strings differ in a single bit. It is well-known that given $\\alpha,\\beta\\in\\{0,1\\}^n$, an $n$-bit Gray code between $\\alpha$ and $\\beta$ exists iff the Hamming distance $d(\\alpha,\\beta)$ of $\\alpha$ and $\\beta$ is odd. We generalize this classical result to $k$ pairwise disjoint pairs $\\alpha_i, \\beta_i\\in\\{0,1\\}^n$: if $d(\\alpha_i,\\beta_i)$ is odd for all $i$ and $k<n$, then the set of all $n$-bit strings can be partitioned into $k$ sequences such that the $i$-th sequence leads from $\\alpha_i$ to $\\beta_i$ and co","authors_text":"Petr Gregor, Tom\\'a\\v{s} Dvo\\v{r}\\'ak, V\\'aclav Koubek","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-03-29T16:16:35Z","title":"Generalized Gray codes with prescribed ends"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.08827","kind":"arxiv","version":1},"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":"sha256:5d486574d702c6fdbfed9910c5781edab9067ad8c4ee49b10b01be6f35217b68","target":"record","created_at":"2026-05-18T00:49:34Z","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":"160561428bafcc85a9f91d16c1fa263e87d14a7607153c79858b07bb0360fbf5","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2016-03-29T16:16:35Z","title_canon_sha256":"69d2cb332548977e2b1f60a294285c8f2830453c100f333204df40844947eed5"},"schema_version":"1.0","source":{"id":"1603.08827","kind":"arxiv","version":1}},"canonical_sha256":"aa43f73818c395a1d3f47b789dd5e8503eb6965bac22cd6f04f34eee633864f1","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aa43f73818c395a1d3f47b789dd5e8503eb6965bac22cd6f04f34eee633864f1","first_computed_at":"2026-05-18T00:49:34.622026Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:49:34.622026Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"LyvxCjD9KhB0713NtAZbTkqnQJV7m4gnbNOXqFwUPvAi+qsQJ7zVG7neHManTbPL8BDDTN5hZsE8hsbfaHd/Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T00:49:34.622608Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.08827","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:5d486574d702c6fdbfed9910c5781edab9067ad8c4ee49b10b01be6f35217b68","sha256:4f7b071cd20807113726bfdde3b9cb7dcb68af8a94d1bfb9a77ab254403c3203"],"state_sha256":"25e8cf1974ac660f71e7341e482db6fe435c9e19fe73ff96eb963daea7f9b21a"}