{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:F5RSNR5A57WPHJ5SFLUBQZQPSI","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":"5f2e6fe2f8ac729f9c61827461ebddca2e4c5d4b292125a3694577b95fe19b1f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2015-03-16T05:44:09Z","title_canon_sha256":"36a31b5fe04dfe472e3216cb0f058aa59bc3993a149945d3e23a999b5fd1ddf7"},"schema_version":"1.0","source":{"id":"1503.04526","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.04526","created_at":"2026-05-18T02:23:24Z"},{"alias_kind":"arxiv_version","alias_value":"1503.04526v1","created_at":"2026-05-18T02:23:24Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.04526","created_at":"2026-05-18T02:23:24Z"},{"alias_kind":"pith_short_12","alias_value":"F5RSNR5A57WP","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"F5RSNR5A57WPHJ5S","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"F5RSNR5A","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:5b258cc84092898b67b0ec1e10f4b5ca3438fc3e71bfcbb32640ce35961f1ee8","target":"graph","created_at":"2026-05-18T02:23:24Z","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":"A hole of a simple connected graph $G$ is a chordless cycle $C_n,$ where $n \\in \\Bbb N, n \\geq 4,$ in the graph $G$. The girth of a simple connected graph $G$ is the smallest cycle in $G$, if any such cycle exists. It can be observed that all such smallest cycles are necessarily chordless. We call the cycle $C_3$ in a given graph $G$ a primitive hole of that graph. We introduce the notion of the primitive hole number of a graph as the number of primitive holes present in that graph. In this paper, we determine the primitive hole number of certain standard graphs. Also, we determine the primiti","authors_text":"Johan Kok, N.K. Sudev","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2015-03-16T05:44:09Z","title":"The Primitive Hole Number of Certain Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.04526","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:ed90343c25c53dc33ce4cbe665a684ce72cbecaf9befdf0d4571f72719907479","target":"record","created_at":"2026-05-18T02:23:24Z","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":"5f2e6fe2f8ac729f9c61827461ebddca2e4c5d4b292125a3694577b95fe19b1f","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/3.0/","primary_cat":"math.CO","submitted_at":"2015-03-16T05:44:09Z","title_canon_sha256":"36a31b5fe04dfe472e3216cb0f058aa59bc3993a149945d3e23a999b5fd1ddf7"},"schema_version":"1.0","source":{"id":"1503.04526","kind":"arxiv","version":1}},"canonical_sha256":"2f6326c7a0efecf3a7b22ae818660f92340cdb4a4496738a6399731c70101985","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f6326c7a0efecf3a7b22ae818660f92340cdb4a4496738a6399731c70101985","first_computed_at":"2026-05-18T02:23:24.736519Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:23:24.736519Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"utuku791UsyOrh/C7BsEW/xqwRVYQR8X5znm3M2XWwmtevGcW1xWdvQy3cBaPP1aVAiCQ5O35q1spVanL9SlAA==","signature_status":"signed_v1","signed_at":"2026-05-18T02:23:24.737155Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.04526","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ed90343c25c53dc33ce4cbe665a684ce72cbecaf9befdf0d4571f72719907479","sha256:5b258cc84092898b67b0ec1e10f4b5ca3438fc3e71bfcbb32640ce35961f1ee8"],"state_sha256":"72ffc21f8b3a0490d5995007e3e07f9d5bfec7e8ed3389c5d97a33bec16f3004"}