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As a consequence, we show that for $r\\ge 2$, if $n\\ge \\frac{1}{r-1}(m+r)(m+2r-1)$ then $K_m\\time"},"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":"1310.2188","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-10-08T15:57:06Z","cross_cats_sorted":["cs.DM"],"title_canon_sha256":"38f42e918d94d6f01d80097658fe2be546635788231128e2528f7d016e7eaf7e","abstract_canon_sha256":"f17de8965882d4aa1af913259ae20f0ba112f0991cb26cb79284e3269ebc34a7"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:10:59.033097Z","signature_b64":"3S3+SrX1Yt2cLpF32BsEgVumrzGSngqii3rsMWuSB2OhM55i/V+/H3SnV/R4PmS+n0FVZ0ZgJ31qlYJMhAceBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"060f8841a32820492745a11685baeff0ace350a4ebbf2f22c0408348e9a2e646","last_reissued_at":"2026-05-18T03:10:59.032327Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:10:59.032327Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On r-equitable chromatic threshold of Kronecker products of complete graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DM"],"primary_cat":"math.CO","authors_text":"Wei Wang, Xin Zhang, Zhidan Yan","submitted_at":"2013-10-08T15:57:06Z","abstract_excerpt":"A graph $G$ is $r$-equitably $k$-colorable if its vertex set can be partitioned into $k$ independent sets, any two of which differ in size by at most $r$. 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