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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1403.7495","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.DM","submitted_at":"2014-03-28T19:13:02Z","cross_cats_sorted":["math.CO"],"title_canon_sha256":"fdde451eed98a910c8a8eca7bab0baf04833c57b89172f5611f0e26c4f7f9c01","abstract_canon_sha256":"271e38936f3da00887f92a0a632ca06416d3f76976e55207ec2ae12adc6fd5d8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:16:02.996283Z","signature_b64":"WPQ/PgRNOPHsjway5o8upxjPO1FKbyJeqPkFzXAkRRUW64zwp2bfVK9leyvoEMI3o9dVXZtM7PpnGNUCl8aMCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"eb571fef50eb7044c7bdd0b1b81eabfc605be21605fd73741e195840ab0d5466","last_reissued_at":"2026-05-18T01:16:02.995543Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:16:02.995543Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"S-Packing Colorings of Cubic Graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"cs.DM","authors_text":"Nicolas Gastineau (GrAMA), Olivier Togni (Le2i)","submitted_at":"2014-03-28T19:13:02Z","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"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.7495","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1403.7495","created_at":"2026-05-18T01:16:02.995664+00:00"},{"alias_kind":"arxiv_version","alias_value":"1403.7495v2","created_at":"2026-05-18T01:16:02.995664+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.7495","created_at":"2026-05-18T01:16:02.995664+00:00"},{"alias_kind":"pith_short_12","alias_value":"5NLR732Q5NYE","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_16","alias_value":"5NLR732Q5NYEJR55","created_at":"2026-05-18T12:28:14.216126+00:00"},{"alias_kind":"pith_short_8","alias_value":"5NLR732Q","created_at":"2026-05-18T12:28:14.216126+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R","json":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R.json","graph_json":"https://pith.science/api/pith-number/5NLR732Q5NYEJR552CY3QHVL7R/graph.json","events_json":"https://pith.science/api/pith-number/5NLR732Q5NYEJR552CY3QHVL7R/events.json","paper":"https://pith.science/paper/5NLR732Q"},"agent_actions":{"view_html":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R","download_json":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R.json","view_paper":"https://pith.science/paper/5NLR732Q","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1403.7495&json=true","fetch_graph":"https://pith.science/api/pith-number/5NLR732Q5NYEJR552CY3QHVL7R/graph.json","fetch_events":"https://pith.science/api/pith-number/5NLR732Q5NYEJR552CY3QHVL7R/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R/action/timestamp_anchor","attest_storage":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R/action/storage_attestation","attest_author":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R/action/author_attestation","sign_citation":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R/action/citation_signature","submit_replication":"https://pith.science/pith/5NLR732Q5NYEJR552CY3QHVL7R/action/replication_record"}},"created_at":"2026-05-18T01:16:02.995664+00:00","updated_at":"2026-05-18T01:16:02.995664+00:00"}