{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2010:F4EKXKXR7OZ27QRTRSA3SS7BGL","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":"8751ed679b49c3dc6dcf0162e99362aa9e1729b0fc405d8538c4a09c35c0a12c","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-18T19:06:34Z","title_canon_sha256":"d0c304e030d8058cc78c6168e663f7e30b6cd432d7a467489c92353d715be86d"},"schema_version":"1.0","source":{"id":"1006.3776","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1006.3776","created_at":"2026-05-18T04:11:18Z"},{"alias_kind":"arxiv_version","alias_value":"1006.3776v1","created_at":"2026-05-18T04:11:18Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.3776","created_at":"2026-05-18T04:11:18Z"},{"alias_kind":"pith_short_12","alias_value":"F4EKXKXR7OZ2","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_16","alias_value":"F4EKXKXR7OZ27QRT","created_at":"2026-05-18T12:26:06Z"},{"alias_kind":"pith_short_8","alias_value":"F4EKXKXR","created_at":"2026-05-18T12:26:06Z"}],"graph_snapshots":[{"event_id":"sha256:5f6627b6573406ab1bad2345f1ad693f5773af9686ab6fe5c1eef7ccd29a489d","target":"graph","created_at":"2026-05-18T04:11:18Z","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":"Let $\\mad(G)$ denote the maximum average degree (over all subgraphs) of $G$ and let $\\chi_i(G)$ denote the injective chromatic number of $G$. We prove that if $\\Delta\\geq 4$ and $\\mad(G)<\\frac{14}5$, then $\\chi_i(G)\\leq\\Delta+2$. When $\\Delta=3$, we show that $\\mad(G)<\\frac{36}{13}$ implies $\\chi_i(G)\\le 5$. In contrast, we give a graph $G$ with $\\Delta=3$, $\\mad(G)=\\frac{36}{13}$, and $\\chi_i(G)=6$.","authors_text":"Daniel W. Cranston, Gexin Yu, Seog-Jin Kim","cross_cats":["cs.DM"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-18T19:06:34Z","title":"Injective colorings of graphs with low average degree"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.3776","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":"sha256:704381656d7db022537756f9e97bf57775b37c4e3b479b5440281f51c2dda380","target":"record","created_at":"2026-05-18T04:11:18Z","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":"8751ed679b49c3dc6dcf0162e99362aa9e1729b0fc405d8538c4a09c35c0a12c","cross_cats_sorted":["cs.DM"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-18T19:06:34Z","title_canon_sha256":"d0c304e030d8058cc78c6168e663f7e30b6cd432d7a467489c92353d715be86d"},"schema_version":"1.0","source":{"id":"1006.3776","kind":"arxiv","version":1}},"canonical_sha256":"2f08abaaf1fbb3afc2338c81b94be132e13b9cd0a1cc7dc80c5a7f923cea81f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2f08abaaf1fbb3afc2338c81b94be132e13b9cd0a1cc7dc80c5a7f923cea81f3","first_computed_at":"2026-05-18T04:11:18.035429Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:11:18.035429Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6ZSBMxp1KW76KWD2ZgyD+58hdRWUu3E8FzBB1Zjair08u2faueM7mH6JK/+577WiTAIY3I5sKq2j8Wpwz+GcDw==","signature_status":"signed_v1","signed_at":"2026-05-18T04:11:18.035940Z","signed_message":"canonical_sha256_bytes"},"source_id":"1006.3776","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:704381656d7db022537756f9e97bf57775b37c4e3b479b5440281f51c2dda380","sha256:5f6627b6573406ab1bad2345f1ad693f5773af9686ab6fe5c1eef7ccd29a489d"],"state_sha256":"bf0ce4bc58a02704e9dbba0758c1a32fd1a740c74a82e8f548fbe46353ef542e"}