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We consider the equivalence number of a line graph, giving improved upper and lower bounds: $\\frac 13 \\log_2\\log_2 \\chi(G) < \\mathit{eq}(L(G)) \\leq 2\\log_2\\log_2 \\chi(G) + 2$. This disproves a recent conjecture that $\\mathit{eq}(L(G))$ is at most three for triangle-free $G$; indeed it can be arbitrarily large.\n  To bound $\\mathit{eq}(L(G))$ we bound the closely-related invariant $\\sigma(G)$, which is the"},"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":"1006.3692","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2010-06-18T13:38:56Z","cross_cats_sorted":[],"title_canon_sha256":"3520409aa57af4992227da55c26e5030ab1a9b7e4284b32e22fc9484f88adcd6","abstract_canon_sha256":"cdc61a700d7ff4dcf84a051ab603095751ed565dba67a8cfc5f6d6b3b1c6bce8"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:28:45.606957Z","signature_b64":"tEpGSHpzunKNFdEO+AIAQV95q0Bd9hE2DGK5SUNESoRPPP5HGQy5r22Alh9MQbwfQZV24hod6gFu//fZeG/IBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"77567f06f554878439236b0178d8a35ca762424d08189de89cebb70f78e18262","last_reissued_at":"2026-05-18T04:28:45.606291Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:28:45.606291Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Covering line graphs with equivalence relations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"A. 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This disproves a recent conjecture that $\\mathit{eq}(L(G))$ is at most three for triangle-free $G$; indeed it can be arbitrarily large.\n  To bound $\\mathit{eq}(L(G))$ we bound the closely-related invariant $\\sigma(G)$, which is the"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1006.3692","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":"1006.3692","created_at":"2026-05-18T04:28:45.606405+00:00"},{"alias_kind":"arxiv_version","alias_value":"1006.3692v1","created_at":"2026-05-18T04:28:45.606405+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1006.3692","created_at":"2026-05-18T04:28:45.606405+00:00"},{"alias_kind":"pith_short_12","alias_value":"O5LH6BXVKSDY","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_16","alias_value":"O5LH6BXVKSDYIOJD","created_at":"2026-05-18T12:26:12.377268+00:00"},{"alias_kind":"pith_short_8","alias_value":"O5LH6BXV","created_at":"2026-05-18T12:26:12.377268+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/O5LH6BXVKSDYIOJDNMAXRWFDLS","json":"https://pith.science/pith/O5LH6BXVKSDYIOJDNMAXRWFDLS.json","graph_json":"https://pith.science/api/pith-number/O5LH6BXVKSDYIOJDNMAXRWFDLS/graph.json","events_json":"https://pith.science/api/pith-number/O5LH6BXVKSDYIOJDNMAXRWFDLS/events.json","paper":"https://pith.science/paper/O5LH6BXV"},"agent_actions":{"view_html":"https://pith.science/pith/O5LH6BXVKSDYIOJDNMAXRWFDLS","download_json":"https://pith.science/pith/O5LH6BXVKSDYIOJDNMAXRWFDLS.json","view_paper":"https://pith.science/paper/O5LH6BXV","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1006.3692&json=true","fetch_graph":"https://pith.science/api/pith-number/O5LH6BXVKSDYIOJDNMAXRWFDLS/graph.json","fetch_events":"https://pith.science/api/pith-number/O5LH6BXVKSDYIOJDNMAXRWFDLS/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/O5LH6BXVKSDYIOJDNMAXRWFDLS/action/timestamp_anchor","attest_storage":"https://pith.science/pith/O5LH6BXVKSDYIOJDNMAXRWFDLS/action/storage_attestation","attest_author":"https://pith.science/pith/O5LH6BXVKSDYIOJDNMAXRWFDLS/action/author_attestation","sign_citation":"https://pith.science/pith/O5LH6BXVKSDYIOJDNMAXRWFDLS/action/citation_signature","submit_replication":"https://pith.science/pith/O5LH6BXVKSDYIOJDNMAXRWFDLS/action/replication_record"}},"created_at":"2026-05-18T04:28:45.606405+00:00","updated_at":"2026-05-18T04:28:45.606405+00:00"}