{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2016:Q67OAIFFAAJXUJCOTCXVZJCKCI","short_pith_number":"pith:Q67OAIFF","schema_version":"1.0","canonical_sha256":"87bee020a500137a244e98af5ca44a121c12371faf4c6814588588a7a099d754","source":{"kind":"arxiv","id":"1608.05573","version":1},"attestation_state":"computed","paper":{"title":"Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo\\v{s}tjan Bre\\v{s}ar, Douglas F. Rall, Kirsti Wash, Sandi Klav\\v{z}ar","submitted_at":"2016-08-19T11:23:13Z","abstract_excerpt":"The packing chromatic number $\\chi_{\\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $\\Pi_1,\\ldots,\\Pi_k$, where $\\Pi_i$, $i\\in [k]$, is an $i$-packing. The following conjecture is posed and studied: if $G$ is a subcubic graph, then $\\chi_{\\rho}(S(G))\\le 5$, where $S(G)$ is the subdivision of $G$. The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if $G$ is a generalized prism of a cycle, then $G$ is $(1,1,2,2)$-colorable if and only if $G$ is not the Petersen graph. The validity o"},"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":"1608.05573","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-08-19T11:23:13Z","cross_cats_sorted":[],"title_canon_sha256":"5bae71e4a58769aa7793dc1bdf52131e439c5fe1d98756947394bd0d8b00fe35","abstract_canon_sha256":"75787fc0633139c5f00c3e51f57a6263c020e032daca342a9507919aad950029"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:08:28.149345Z","signature_b64":"4UntKTjjJumlTskVtDkc5j/FmVlHAZ01A4dGaK4KC5Prvr6xGfJ+s4gsoXvh+bHQ2irrdiQuwaiUTaroICc4BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"87bee020a500137a244e98af5ca44a121c12371faf4c6814588588a7a099d754","last_reissued_at":"2026-05-18T01:08:28.148780Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:08:28.148780Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Packing chromatic number, $(1,1,2,2)$-colorings, and characterizing the Petersen graph","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bo\\v{s}tjan Bre\\v{s}ar, Douglas F. Rall, Kirsti Wash, Sandi Klav\\v{z}ar","submitted_at":"2016-08-19T11:23:13Z","abstract_excerpt":"The packing chromatic number $\\chi_{\\rho}(G)$ of a graph $G$ is the smallest integer $k$ such that the vertex set of $G$ can be partitioned into sets $\\Pi_1,\\ldots,\\Pi_k$, where $\\Pi_i$, $i\\in [k]$, is an $i$-packing. The following conjecture is posed and studied: if $G$ is a subcubic graph, then $\\chi_{\\rho}(S(G))\\le 5$, where $S(G)$ is the subdivision of $G$. The conjecture is proved for all generalized prisms of cycles. To get this result it is proved that if $G$ is a generalized prism of a cycle, then $G$ is $(1,1,2,2)$-colorable if and only if $G$ is not the Petersen graph. The validity o"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.05573","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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1608.05573","created_at":"2026-05-18T01:08:28.148865+00:00"},{"alias_kind":"arxiv_version","alias_value":"1608.05573v1","created_at":"2026-05-18T01:08:28.148865+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.05573","created_at":"2026-05-18T01:08:28.148865+00:00"},{"alias_kind":"pith_short_12","alias_value":"Q67OAIFFAAJX","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_16","alias_value":"Q67OAIFFAAJXUJCO","created_at":"2026-05-18T12:30:39.010887+00:00"},{"alias_kind":"pith_short_8","alias_value":"Q67OAIFF","created_at":"2026-05-18T12:30:39.010887+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/Q67OAIFFAAJXUJCOTCXVZJCKCI","json":"https://pith.science/pith/Q67OAIFFAAJXUJCOTCXVZJCKCI.json","graph_json":"https://pith.science/api/pith-number/Q67OAIFFAAJXUJCOTCXVZJCKCI/graph.json","events_json":"https://pith.science/api/pith-number/Q67OAIFFAAJXUJCOTCXVZJCKCI/events.json","paper":"https://pith.science/paper/Q67OAIFF"},"agent_actions":{"view_html":"https://pith.science/pith/Q67OAIFFAAJXUJCOTCXVZJCKCI","download_json":"https://pith.science/pith/Q67OAIFFAAJXUJCOTCXVZJCKCI.json","view_paper":"https://pith.science/paper/Q67OAIFF","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1608.05573&json=true","fetch_graph":"https://pith.science/api/pith-number/Q67OAIFFAAJXUJCOTCXVZJCKCI/graph.json","fetch_events":"https://pith.science/api/pith-number/Q67OAIFFAAJXUJCOTCXVZJCKCI/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/Q67OAIFFAAJXUJCOTCXVZJCKCI/action/timestamp_anchor","attest_storage":"https://pith.science/pith/Q67OAIFFAAJXUJCOTCXVZJCKCI/action/storage_attestation","attest_author":"https://pith.science/pith/Q67OAIFFAAJXUJCOTCXVZJCKCI/action/author_attestation","sign_citation":"https://pith.science/pith/Q67OAIFFAAJXUJCOTCXVZJCKCI/action/citation_signature","submit_replication":"https://pith.science/pith/Q67OAIFFAAJXUJCOTCXVZJCKCI/action/replication_record"}},"created_at":"2026-05-18T01:08:28.148865+00:00","updated_at":"2026-05-18T01:08:28.148865+00:00"}