{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2010:NY5IVTLNYFLOOS6LGBHHOJXVST","short_pith_number":"pith:NY5IVTLN","canonical_record":{"source":{"id":"1008.2583","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-16T06:59:15Z","cross_cats_sorted":[],"title_canon_sha256":"ded55a87b16567e669c27c7960c07334ae16d801024e635ef8b69441a6c50bb9","abstract_canon_sha256":"d6f442a153697046dcf79f19cb26bac5e11bcc435570a8630ca86656e4d0516c"},"schema_version":"1.0"},"canonical_sha256":"6e3a8acd6dc156e74bcb304e7726f594edb15c5b2f1131451fd8caa771807ffd","source":{"kind":"arxiv","id":"1008.2583","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.2583","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"arxiv_version","alias_value":"1008.2583v2","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.2583","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"pith_short_12","alias_value":"NY5IVTLNYFLO","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"NY5IVTLNYFLOOS6L","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"NY5IVTLN","created_at":"2026-05-18T12:26:12Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2010:NY5IVTLNYFLOOS6LGBHHOJXVST","target":"record","payload":{"canonical_record":{"source":{"id":"1008.2583","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-16T06:59:15Z","cross_cats_sorted":[],"title_canon_sha256":"ded55a87b16567e669c27c7960c07334ae16d801024e635ef8b69441a6c50bb9","abstract_canon_sha256":"d6f442a153697046dcf79f19cb26bac5e11bcc435570a8630ca86656e4d0516c"},"schema_version":"1.0"},"canonical_sha256":"6e3a8acd6dc156e74bcb304e7726f594edb15c5b2f1131451fd8caa771807ffd","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:27:47.536523Z","signature_b64":"cWae7pndHtSpes8Y0n6VyrRP58Bx3B/arXjO3ktGLiqQZuTAknSt859dcmpPTrUeSB5RE62Hc7tbObipSc1mBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6e3a8acd6dc156e74bcb304e7726f594edb15c5b2f1131451fd8caa771807ffd","last_reissued_at":"2026-05-18T04:27:47.535924Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:27:47.535924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1008.2583","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":"2026-05-18T04:27:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iGZNppex2Qkue2vr+jgory3PMQh0O7pZ86GT/mkqotxsntrXa+LiF+w/WBOlFNscWkxYSYUc7JyAaNrSHFKpCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:19:34.816936Z"},"content_sha256":"8e12b38249dad95d8f3059b5b85e1cac220cfc0aa53e713795058876fe7795cd","schema_version":"1.0","event_id":"sha256:8e12b38249dad95d8f3059b5b85e1cac220cfc0aa53e713795058876fe7795cd"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2010:NY5IVTLNYFLOOS6LGBHHOJXVST","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Independence number of generalized Petersen graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. Azadi, J. Ebrahimi B, Nazli Besharati","submitted_at":"2010-08-16T06:59:15Z","abstract_excerpt":"Determining the size of a maximum independent set of a graph $G$, denoted by $\\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\\alpha (G)$ for special classes of graphs.\n  This paper is aimed toward studying this problem for the class of generalized Petersen graphs. We find new upper and lower bounds and some exact values for $\\alpha(P(n,k))$. With a computer program we have obtained exact values for each $n<78$. In \\cite{MR2381433} it is conjectured that $\\beta(P(n, k)) \\leq n + \\lceil\\frac{n}{5}\\rceil $, for all $n$ and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2583","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":"2026-05-18T04:27:47Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"X9OoGQhPf31MBg0BF/IblYXWEjbCJzetpwWYj4LLGntP6s047VhkYR21AZxJPdhYnJqn1cF9VyruFmdaCZcDAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-27T02:19:34.817265Z"},"content_sha256":"0641ce5fdb4a43f6e6e031cb17b61a528f80e92b65582fbc2fa61f177b2b7cb0","schema_version":"1.0","event_id":"sha256:0641ce5fdb4a43f6e6e031cb17b61a528f80e92b65582fbc2fa61f177b2b7cb0"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/NY5IVTLNYFLOOS6LGBHHOJXVST/bundle.json","state_url":"https://pith.science/pith/NY5IVTLNYFLOOS6LGBHHOJXVST/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/NY5IVTLNYFLOOS6LGBHHOJXVST/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":"2026-06-27T02:19:34Z","links":{"resolver":"https://pith.science/pith/NY5IVTLNYFLOOS6LGBHHOJXVST","bundle":"https://pith.science/pith/NY5IVTLNYFLOOS6LGBHHOJXVST/bundle.json","state":"https://pith.science/pith/NY5IVTLNYFLOOS6LGBHHOJXVST/state.json","well_known_bundle":"https://pith.science/.well-known/pith/NY5IVTLNYFLOOS6LGBHHOJXVST/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:NY5IVTLNYFLOOS6LGBHHOJXVST","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":"d6f442a153697046dcf79f19cb26bac5e11bcc435570a8630ca86656e4d0516c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-16T06:59:15Z","title_canon_sha256":"ded55a87b16567e669c27c7960c07334ae16d801024e635ef8b69441a6c50bb9"},"schema_version":"1.0","source":{"id":"1008.2583","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1008.2583","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"arxiv_version","alias_value":"1008.2583v2","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1008.2583","created_at":"2026-05-18T04:27:47Z"},{"alias_kind":"pith_short_12","alias_value":"NY5IVTLNYFLO","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_16","alias_value":"NY5IVTLNYFLOOS6L","created_at":"2026-05-18T12:26:12Z"},{"alias_kind":"pith_short_8","alias_value":"NY5IVTLN","created_at":"2026-05-18T12:26:12Z"}],"graph_snapshots":[{"event_id":"sha256:0641ce5fdb4a43f6e6e031cb17b61a528f80e92b65582fbc2fa61f177b2b7cb0","target":"graph","created_at":"2026-05-18T04:27:47Z","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":"Determining the size of a maximum independent set of a graph $G$, denoted by $\\alpha(G)$, is an NP-hard problem. Therefore, many attempts are made to find upper and lower bounds, or exact values of $\\alpha (G)$ for special classes of graphs.\n  This paper is aimed toward studying this problem for the class of generalized Petersen graphs. We find new upper and lower bounds and some exact values for $\\alpha(P(n,k))$. With a computer program we have obtained exact values for each $n<78$. In \\cite{MR2381433} it is conjectured that $\\beta(P(n, k)) \\leq n + \\lceil\\frac{n}{5}\\rceil $, for all $n$ and ","authors_text":"A. Azadi, J. Ebrahimi B, Nazli Besharati","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-16T06:59:15Z","title":"Independence number of generalized Petersen graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1008.2583","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":"sha256:8e12b38249dad95d8f3059b5b85e1cac220cfc0aa53e713795058876fe7795cd","target":"record","created_at":"2026-05-18T04:27:47Z","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":"d6f442a153697046dcf79f19cb26bac5e11bcc435570a8630ca86656e4d0516c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-08-16T06:59:15Z","title_canon_sha256":"ded55a87b16567e669c27c7960c07334ae16d801024e635ef8b69441a6c50bb9"},"schema_version":"1.0","source":{"id":"1008.2583","kind":"arxiv","version":2}},"canonical_sha256":"6e3a8acd6dc156e74bcb304e7726f594edb15c5b2f1131451fd8caa771807ffd","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6e3a8acd6dc156e74bcb304e7726f594edb15c5b2f1131451fd8caa771807ffd","first_computed_at":"2026-05-18T04:27:47.535924Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:27:47.535924Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cWae7pndHtSpes8Y0n6VyrRP58Bx3B/arXjO3ktGLiqQZuTAknSt859dcmpPTrUeSB5RE62Hc7tbObipSc1mBg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:27:47.536523Z","signed_message":"canonical_sha256_bytes"},"source_id":"1008.2583","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:8e12b38249dad95d8f3059b5b85e1cac220cfc0aa53e713795058876fe7795cd","sha256:0641ce5fdb4a43f6e6e031cb17b61a528f80e92b65582fbc2fa61f177b2b7cb0"],"state_sha256":"282c7f818aab39d0acb1db5b7be975432d172a658c7c6a20152566d41fcad457"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"IR4cLkTQji22R0NNofeq9e217IPwCaxaYGW6DNkwXGwYhgjsFh7DW8joIdr7LgZKE0W51QauGfGOZbrzaDKVAA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-27T02:19:34.819038Z","bundle_sha256":"9584762fede4c1c3748da50a05dff9f1e64bbbd564e1eff52307216461791438"}}