{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:WRV237RGX5S2YQZQ3YTDNGVDNB","short_pith_number":"pith:WRV237RG","canonical_record":{"source":{"id":"1507.03160","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-11T22:05:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5abc508ba3af099ee05c558946b40c0e7fed3f06b1843c39a19f325122ddec5d","abstract_canon_sha256":"afa92aace456ddb4554ea087913668682665824e6e566ebff7063d4e8ad91254"},"schema_version":"1.0"},"canonical_sha256":"b46badfe26bf65ac4330de26369aa36842a78b07d1236bf15d0aa44062c98bd4","source":{"kind":"arxiv","id":"1507.03160","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.03160","created_at":"2026-05-18T01:36:46Z"},{"alias_kind":"arxiv_version","alias_value":"1507.03160v1","created_at":"2026-05-18T01:36:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03160","created_at":"2026-05-18T01:36:46Z"},{"alias_kind":"pith_short_12","alias_value":"WRV237RGX5S2","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WRV237RGX5S2YQZQ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WRV237RG","created_at":"2026-05-18T12:29:47Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:WRV237RGX5S2YQZQ3YTDNGVDNB","target":"record","payload":{"canonical_record":{"source":{"id":"1507.03160","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-11T22:05:52Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"5abc508ba3af099ee05c558946b40c0e7fed3f06b1843c39a19f325122ddec5d","abstract_canon_sha256":"afa92aace456ddb4554ea087913668682665824e6e566ebff7063d4e8ad91254"},"schema_version":"1.0"},"canonical_sha256":"b46badfe26bf65ac4330de26369aa36842a78b07d1236bf15d0aa44062c98bd4","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:36:46.451576Z","signature_b64":"5NKAwbRLhB23tRwUu0rNwY4crLpfvQc1oikpBz6MCPg1Dqy6D/QM/6wK7MdrOjmyfCd3H2dGiuX0nny37NKOCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b46badfe26bf65ac4330de26369aa36842a78b07d1236bf15d0aa44062c98bd4","last_reissued_at":"2026-05-18T01:36:46.451124Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:36:46.451124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1507.03160","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:36:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"9OdD8ND7dmq5XUZq5RVWyYclpqK2C/G2ILEjSeNbq5piCZdN9pP8DDC3FY/qAfxX5gTahp8LP/gKqV+47HbvBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T18:19:50.383291Z"},"content_sha256":"fb9885fe05340e9268f5348af7da18fa61886c86f275f50513469de7f053934a","schema_version":"1.0","event_id":"sha256:fb9885fe05340e9268f5348af7da18fa61886c86f275f50513469de7f053934a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:WRV237RGX5S2YQZQ3YTDNGVDNB","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Strong $(r,p)$ Cover for Hypergraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Sudebkumar Prasant Pal, Tapas Kumar Mishra","submitted_at":"2015-07-11T22:05:52Z","abstract_excerpt":"We introduce the notion of the { \\it strong $(r,p)$ cover} number $\\chi^c(G,k,r,p)$ for $k$-uniform hypergraphs $G(V,E)$, where $\\chi^c(G,k,r,p)$ denotes the minimum number of $r$-colorings of vertices in $V$ such that each hyperedge in $E$ contains at least $min(p,k)$ vertices of distinct colors in at least one of the $\\chi^c(G,k,r,p)$ $r$-colorings. We derive the exact values of $\\chi^c(K_n^k,k,r,p)$ for small values of $n$, $k$, $r$ and $p$, where $K_n^k$ denotes the complete $k$-uniform hypergraph of $n$ vertices. We study the variation of $\\chi^c(G,k,r,p)$ with respect to changes in $k$, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03160","kind":"arxiv","version":1},"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":"2026-05-18T01:36:46Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tHAAyQSCuyU6vPQv/OLobe3tfMklHzI4jsT0ZvTbkZlJQYPUAXsy+iyP9PxM8nGDElOHiv4op0t52IcZ80h0Bw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-29T18:19:50.383654Z"},"content_sha256":"719263d65d40627ac405817cd18ae291f7a5513e0b6660471f13f25d4ea99ed4","schema_version":"1.0","event_id":"sha256:719263d65d40627ac405817cd18ae291f7a5513e0b6660471f13f25d4ea99ed4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/WRV237RGX5S2YQZQ3YTDNGVDNB/bundle.json","state_url":"https://pith.science/pith/WRV237RGX5S2YQZQ3YTDNGVDNB/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/WRV237RGX5S2YQZQ3YTDNGVDNB/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":"2026-06-29T18:19:50Z","links":{"resolver":"https://pith.science/pith/WRV237RGX5S2YQZQ3YTDNGVDNB","bundle":"https://pith.science/pith/WRV237RGX5S2YQZQ3YTDNGVDNB/bundle.json","state":"https://pith.science/pith/WRV237RGX5S2YQZQ3YTDNGVDNB/state.json","well_known_bundle":"https://pith.science/.well-known/pith/WRV237RGX5S2YQZQ3YTDNGVDNB/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:WRV237RGX5S2YQZQ3YTDNGVDNB","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":"afa92aace456ddb4554ea087913668682665824e6e566ebff7063d4e8ad91254","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-11T22:05:52Z","title_canon_sha256":"5abc508ba3af099ee05c558946b40c0e7fed3f06b1843c39a19f325122ddec5d"},"schema_version":"1.0","source":{"id":"1507.03160","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1507.03160","created_at":"2026-05-18T01:36:46Z"},{"alias_kind":"arxiv_version","alias_value":"1507.03160v1","created_at":"2026-05-18T01:36:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1507.03160","created_at":"2026-05-18T01:36:46Z"},{"alias_kind":"pith_short_12","alias_value":"WRV237RGX5S2","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_16","alias_value":"WRV237RGX5S2YQZQ","created_at":"2026-05-18T12:29:47Z"},{"alias_kind":"pith_short_8","alias_value":"WRV237RG","created_at":"2026-05-18T12:29:47Z"}],"graph_snapshots":[{"event_id":"sha256:719263d65d40627ac405817cd18ae291f7a5513e0b6660471f13f25d4ea99ed4","target":"graph","created_at":"2026-05-18T01:36:46Z","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":"We introduce the notion of the { \\it strong $(r,p)$ cover} number $\\chi^c(G,k,r,p)$ for $k$-uniform hypergraphs $G(V,E)$, where $\\chi^c(G,k,r,p)$ denotes the minimum number of $r$-colorings of vertices in $V$ such that each hyperedge in $E$ contains at least $min(p,k)$ vertices of distinct colors in at least one of the $\\chi^c(G,k,r,p)$ $r$-colorings. We derive the exact values of $\\chi^c(K_n^k,k,r,p)$ for small values of $n$, $k$, $r$ and $p$, where $K_n^k$ denotes the complete $k$-uniform hypergraph of $n$ vertices. We study the variation of $\\chi^c(G,k,r,p)$ with respect to changes in $k$, ","authors_text":"Sudebkumar Prasant Pal, Tapas Kumar Mishra","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-11T22:05:52Z","title":"Strong $(r,p)$ Cover for Hypergraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1507.03160","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:fb9885fe05340e9268f5348af7da18fa61886c86f275f50513469de7f053934a","target":"record","created_at":"2026-05-18T01:36:46Z","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":"afa92aace456ddb4554ea087913668682665824e6e566ebff7063d4e8ad91254","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2015-07-11T22:05:52Z","title_canon_sha256":"5abc508ba3af099ee05c558946b40c0e7fed3f06b1843c39a19f325122ddec5d"},"schema_version":"1.0","source":{"id":"1507.03160","kind":"arxiv","version":1}},"canonical_sha256":"b46badfe26bf65ac4330de26369aa36842a78b07d1236bf15d0aa44062c98bd4","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"b46badfe26bf65ac4330de26369aa36842a78b07d1236bf15d0aa44062c98bd4","first_computed_at":"2026-05-18T01:36:46.451124Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:36:46.451124Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"5NKAwbRLhB23tRwUu0rNwY4crLpfvQc1oikpBz6MCPg1Dqy6D/QM/6wK7MdrOjmyfCd3H2dGiuX0nny37NKOCw==","signature_status":"signed_v1","signed_at":"2026-05-18T01:36:46.451576Z","signed_message":"canonical_sha256_bytes"},"source_id":"1507.03160","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:fb9885fe05340e9268f5348af7da18fa61886c86f275f50513469de7f053934a","sha256:719263d65d40627ac405817cd18ae291f7a5513e0b6660471f13f25d4ea99ed4"],"state_sha256":"a0117722caa8e159a0844f5c24e752526bd7a83e207da9c13bcc6c4998c051af"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6snutdEcNgnoHiU+1YeMjX0aG1CSJ+VdTyfpWQMuwE5qdxrdRpZ5Cyz3PvoE2Uj/4tHeQzKH56yRnDUrcD+KCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-29T18:19:50.386129Z","bundle_sha256":"39a002e02309f29c5b0fdb105c24fced939c46b7618f21bf1965e43a78b42e45"}}