{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:JHOZN22LSP4N6VMW2KPYKPXFLP","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":"78e1189af878b31179ae23f8ad688d3ec98b2293a5b171db8e2d354e3d081105","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-24T19:25:40Z","title_canon_sha256":"39f53afbbd720541bfb71adb300a0421e20b354e2427547f66766e5bb1949f70"},"schema_version":"1.0","source":{"id":"1205.5537","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1205.5537","created_at":"2026-05-18T03:55:00Z"},{"alias_kind":"arxiv_version","alias_value":"1205.5537v1","created_at":"2026-05-18T03:55:00Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1205.5537","created_at":"2026-05-18T03:55:00Z"},{"alias_kind":"pith_short_12","alias_value":"JHOZN22LSP4N","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_16","alias_value":"JHOZN22LSP4N6VMW","created_at":"2026-05-18T12:27:11Z"},{"alias_kind":"pith_short_8","alias_value":"JHOZN22L","created_at":"2026-05-18T12:27:11Z"}],"graph_snapshots":[{"event_id":"sha256:9e04d1be894654e139a62d8adca7d264820be23fde0268a166d00f6999fc366f","target":"graph","created_at":"2026-05-18T03:55:00Z","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 $\\gamma(C_m\\Box C_n)$ be the domination number of the Cartesian product of directed cycles $C_m$ and $C_n$ for $m,n\\geq2$. Shaheen [] and Liu and al.[ ], [ ] determined the value of $\\gamma(C_m\\Box C_n)$ when $m \\leq 6$ and when both $m$ and $n$ $\\equiv 0$ $(mod\\: 3)$. In this article we give, in general, the value of $\\gamma(C_m\\Box C_n)$ when $m\\equiv 2$ $(mod\\: 3)$ and improve the known lower bound for most of the remaining cases. We also disprove the conjectured formula for the case $m$ $\\equiv 0$ $(mod\\: 3)$ appearing in \\cite{}","authors_text":"Michel Mollard (IF)","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-24T19:25:40Z","title":"On domination of Cartesian product of directed cycles"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1205.5537","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":"sha256:377d72f27172fe63723d9134fe543146a199165921256c869723a9db32b9ea5f","target":"record","created_at":"2026-05-18T03:55:00Z","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":"78e1189af878b31179ae23f8ad688d3ec98b2293a5b171db8e2d354e3d081105","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-05-24T19:25:40Z","title_canon_sha256":"39f53afbbd720541bfb71adb300a0421e20b354e2427547f66766e5bb1949f70"},"schema_version":"1.0","source":{"id":"1205.5537","kind":"arxiv","version":1}},"canonical_sha256":"49dd96eb4b93f8df5596d29f853ee55bcbd19f99e41b7835430bce158b2ab48c","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"49dd96eb4b93f8df5596d29f853ee55bcbd19f99e41b7835430bce158b2ab48c","first_computed_at":"2026-05-18T03:55:00.861863Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:55:00.861863Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"QY0KCRd6W8zrWeb7kY41Ebku+B0n7Gwe2pP63obEnrpWQasoKNpo4oAyWFdP2a+VvgIDZPc7Xuazu48ElZJHAg==","signature_status":"signed_v1","signed_at":"2026-05-18T03:55:00.862441Z","signed_message":"canonical_sha256_bytes"},"source_id":"1205.5537","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:377d72f27172fe63723d9134fe543146a199165921256c869723a9db32b9ea5f","sha256:9e04d1be894654e139a62d8adca7d264820be23fde0268a166d00f6999fc366f"],"state_sha256":"13c6712c982eba66776f8eacf1109ca9bdf30e8169a122feda578fca0bf16890"}