{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:3R556BXGQKDMID4UQ5QBB5CGZC","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":"7571bcd83086ed2fe03eb690133480d802c95041b4cbb182f7d51606bf351c08","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-21T01:29:31Z","title_canon_sha256":"e945a8de85f505a724a4852aed3cec119a245475c2a2dc2d1ea9ea402853b3d6"},"schema_version":"1.0","source":{"id":"1303.5136","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1303.5136","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"arxiv_version","alias_value":"1303.5136v3","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1303.5136","created_at":"2026-05-18T01:35:53Z"},{"alias_kind":"pith_short_12","alias_value":"3R556BXGQKDM","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_16","alias_value":"3R556BXGQKDMID4U","created_at":"2026-05-18T12:27:32Z"},{"alias_kind":"pith_short_8","alias_value":"3R556BXG","created_at":"2026-05-18T12:27:32Z"}],"graph_snapshots":[{"event_id":"sha256:75beef995e640324988f587857130244a876f98503f90fcef5d246b4d4d9d5fc","target":"graph","created_at":"2026-05-18T01:35:53Z","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 determine the list chromatic number of the square of a graph $\\chil(G^2)$ in terms of its maximum degree $\\Delta$ when its maximum average degree, denoted $\\mad(G)$, is sufficiently small. For $\\Delta\\ge 6$, if $\\mad(G)<2+\\frac{4\\Delta-8}{5\\Delta+2}$, then $\\chil(G^2)=\\Delta+1$. In particular, if $G$ is planar with girth $g\\ge 7+\\frac{12}{\\Delta-2}$, then $\\chil(G^2)=\\Delta+1$. Under the same conditions, $\\chil^i(G)=\\Delta$, where $\\chil^i$ is the list injective chromatic number.","authors_text":"Daniel W. Cranston, Riste \\v{S}krekovski","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-21T01:29:31Z","title":"Sufficient sparseness conditions for G^2 to be (\\Delta+1)-choosable, when \\Delta\\ge5"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1303.5136","kind":"arxiv","version":3},"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":"sha256:1da9327e3ae03bace3a72854e4d57f4a78f972aa9c2d2bc99ecdbe8a0a6be71f","target":"record","created_at":"2026-05-18T01:35:53Z","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":"7571bcd83086ed2fe03eb690133480d802c95041b4cbb182f7d51606bf351c08","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-03-21T01:29:31Z","title_canon_sha256":"e945a8de85f505a724a4852aed3cec119a245475c2a2dc2d1ea9ea402853b3d6"},"schema_version":"1.0","source":{"id":"1303.5136","kind":"arxiv","version":3}},"canonical_sha256":"dc7bdf06e68286c40f94876010f446c8a977740026850161f3fb3f057227972b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"dc7bdf06e68286c40f94876010f446c8a977740026850161f3fb3f057227972b","first_computed_at":"2026-05-18T01:35:53.001627Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:35:53.001627Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"i/Z2h8xrZrkNYwcZApirkaNFUuZSZEogxdL4pu3uw7/wUaE73uVWds1O/RLgAyJkKMKUoxK+tHOOZVsL80EpCg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:35:53.002332Z","signed_message":"canonical_sha256_bytes"},"source_id":"1303.5136","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:1da9327e3ae03bace3a72854e4d57f4a78f972aa9c2d2bc99ecdbe8a0a6be71f","sha256:75beef995e640324988f587857130244a876f98503f90fcef5d246b4d4d9d5fc"],"state_sha256":"f3f8940a7c6072b0633281e02fbcdc8909eb497c5f4f8dae1c18bfae482bfc32"}