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[J Graph Theory 66 (2011), 89--97] and Jamison [Discrete Math. 74 (1989), 99--115] in the following values: if $ x\\geq 2 $ is an integer and $m\\in \\{4x^2-x,\\dots,4x^2+3x-3\\}$ then $\\psi(m) \\leq 2x(m-x-1)$.\n  On the other hand, if $ q $ is even and there exists $ \\Pi_q $ we give a complete ed"},"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":"1507.08338","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-07-29T22:45:59Z","cross_cats_sorted":[],"title_canon_sha256":"6f92baf4a81d64a97d9904cc4e2b8eed4222aaa4d24044a8db73f53fc318d04e","abstract_canon_sha256":"b6f75a4a7eb4196d558e0c496b162702ff6c5917f115d51e3e41302353f79519"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:04:34.402774Z","signature_b64":"HOJwP3Q/yFVwobugJ7GdXjr7gt0wj9mI0EpOf/jzXYzK+mrRHQkOeWoGoH2V1MxUTD7IKJKwN5kkarWrTxIjAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6ff43de4732878a787969ce5b8185a9200664df85663c88f26ee495966a7de29","last_reissued_at":"2026-05-18T00:04:34.402111Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:04:34.402111Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the pseudoachromatic index of the complete graph III","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"C. 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