{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:5NLR732Q5NYEJR552CY3QHVL7R","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":"271e38936f3da00887f92a0a632ca06416d3f76976e55207ec2ae12adc6fd5d8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-03-28T19:13:02Z","title_canon_sha256":"fdde451eed98a910c8a8eca7bab0baf04833c57b89172f5611f0e26c4f7f9c01"},"schema_version":"1.0","source":{"id":"1403.7495","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.7495","created_at":"2026-05-18T01:16:02Z"},{"alias_kind":"arxiv_version","alias_value":"1403.7495v2","created_at":"2026-05-18T01:16:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.7495","created_at":"2026-05-18T01:16:02Z"},{"alias_kind":"pith_short_12","alias_value":"5NLR732Q5NYE","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_16","alias_value":"5NLR732Q5NYEJR55","created_at":"2026-05-18T12:28:14Z"},{"alias_kind":"pith_short_8","alias_value":"5NLR732Q","created_at":"2026-05-18T12:28:14Z"}],"graph_snapshots":[{"event_id":"sha256:1f1eba0f6cf376f28c5fbba48f23afa8e5bbd7721a8ff985f14ddd60d2ce2080","target":"graph","created_at":"2026-05-18T01:16:02Z","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":"Given a non-decreasing sequence $S=(s\\_1,s\\_2, \\ldots, s\\_k)$ of positive integers, an {\\em $S$-packing coloring} of a graph $G$ is a mapping $c$ from $V(G)$ to $\\{s\\_1,s\\_2, \\ldots, s\\_k\\}$ such that any two vertices with color $s\\_i$ are at mutual distance greater than $s\\_i$, $1\\le i\\le k$. This paper studies $S$-packing colorings of (sub)cubic graphs. We prove that subcubic graphs are $(1,2,2,2,2,2,2)$-packing colorable and $(1,1,2,2,3)$-packing colorable. For subdivisions of subcubic graphs we derive sharper bounds, and we provide an example of a cubic graph of order $38$ which is not $(1","authors_text":"Nicolas Gastineau (GrAMA), Olivier Togni (Le2i)","cross_cats":["math.CO"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-03-28T19:13:02Z","title":"S-Packing Colorings of Cubic Graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7495","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:d56df5a640299d375146fb4008bd73bec8eb599343a6609a9d44b707f2bc6e24","target":"record","created_at":"2026-05-18T01:16:02Z","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":"271e38936f3da00887f92a0a632ca06416d3f76976e55207ec2ae12adc6fd5d8","cross_cats_sorted":["math.CO"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-03-28T19:13:02Z","title_canon_sha256":"fdde451eed98a910c8a8eca7bab0baf04833c57b89172f5611f0e26c4f7f9c01"},"schema_version":"1.0","source":{"id":"1403.7495","kind":"arxiv","version":2}},"canonical_sha256":"eb571fef50eb7044c7bdd0b1b81eabfc605be21605fd73741e195840ab0d5466","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"eb571fef50eb7044c7bdd0b1b81eabfc605be21605fd73741e195840ab0d5466","first_computed_at":"2026-05-18T01:16:02.995543Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:16:02.995543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WPQ/PgRNOPHsjway5o8upxjPO1FKbyJeqPkFzXAkRRUW64zwp2bfVK9leyvoEMI3o9dVXZtM7PpnGNUCl8aMCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:16:02.996283Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.7495","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d56df5a640299d375146fb4008bd73bec8eb599343a6609a9d44b707f2bc6e24","sha256:1f1eba0f6cf376f28c5fbba48f23afa8e5bbd7721a8ff985f14ddd60d2ce2080"],"state_sha256":"b373481b4bbe62b6007e09ca842c5ea2d5915d670465736943e179df09b05624"}