{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:4NXSZL3N2SSURI65CU5GBJKHUT","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":"a751aa2e9261e63e2d461b9dbd2f632010a31e4c05cad7f9e3161fa6588fec4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-08T13:17:15Z","title_canon_sha256":"847833e9343e80af480a10e07563055fa50b2b77e9bf023fafa2e20688efc270"},"schema_version":"1.0","source":{"id":"1501.01833","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1501.01833","created_at":"2026-05-18T02:29:46Z"},{"alias_kind":"arxiv_version","alias_value":"1501.01833v1","created_at":"2026-05-18T02:29:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1501.01833","created_at":"2026-05-18T02:29:46Z"},{"alias_kind":"pith_short_12","alias_value":"4NXSZL3N2SSU","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_16","alias_value":"4NXSZL3N2SSURI65","created_at":"2026-05-18T12:29:05Z"},{"alias_kind":"pith_short_8","alias_value":"4NXSZL3N","created_at":"2026-05-18T12:29:05Z"}],"graph_snapshots":[{"event_id":"sha256:7ac459822fc108d94e740e5af0cc55d24c27d5836043397c144747e34c5d37d7","target":"graph","created_at":"2026-05-18T02:29:46Z","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":"The k-limited packing number, $L_k(G)$, of a graph $G$, introduced by Gallant, Gunther, Hartnell, and Rall, is the maximum cardinality of a set $X$ of vertices of $G$ such that every vertex of $G$ has at most $k$ elements of $X$ in its closed neighbourhood. The main aim in this paper is to prove the best-possible result that if $G$ is a cubic graph, then $L_2(G) \\geq |V (G)|/3$, improving the previous lower bound given by Gallant, \\emph{et al.}\n  In addition, we construct an infinite family of graphs to show that lower bounds given by Gagarin and Zverovich are asymptotically best-possible, up ","authors_text":"B\\'ela Bollob\\'as, Karen Gunderson, Paul N. Balister","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-08T13:17:15Z","title":"Limited packings of closed neighbourhoods in graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1501.01833","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":"sha256:37e9f05fa62a68474e087e1c7567c23be17c1e6ad197d29aa4927ce86b67ea6d","target":"record","created_at":"2026-05-18T02:29:46Z","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":"a751aa2e9261e63e2d461b9dbd2f632010a31e4c05cad7f9e3161fa6588fec4a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-01-08T13:17:15Z","title_canon_sha256":"847833e9343e80af480a10e07563055fa50b2b77e9bf023fafa2e20688efc270"},"schema_version":"1.0","source":{"id":"1501.01833","kind":"arxiv","version":1}},"canonical_sha256":"e36f2caf6dd4a548a3dd153a60a547a4e6054f69a3680475c8e7460c04f7837b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"e36f2caf6dd4a548a3dd153a60a547a4e6054f69a3680475c8e7460c04f7837b","first_computed_at":"2026-05-18T02:29:46.311431Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:29:46.311431Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"cwnjrlbkiIWqK3xjaRtr9dCBtSHLqomrBKSoZrWUdQ5clrRJgWJPMhQjWjkuyPjx+GPSNeXCT0tcUcZFnNv0Cg==","signature_status":"signed_v1","signed_at":"2026-05-18T02:29:46.311828Z","signed_message":"canonical_sha256_bytes"},"source_id":"1501.01833","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:37e9f05fa62a68474e087e1c7567c23be17c1e6ad197d29aa4927ce86b67ea6d","sha256:7ac459822fc108d94e740e5af0cc55d24c27d5836043397c144747e34c5d37d7"],"state_sha256":"6d42f3bfac10d2a3404d8933a47023edc513057f6b876406928a1d15daf18304"}