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Let $G\\times H$ denote the direct product of graphs $G$ and $H$. For $n\\ge m\\ge 2$ we prove that $\\chi_=^*(K_{m} \\times K_n)$ equals $\\lceil\\frac{mn}{m+1}\\rceil$ if $n\\equiv 2,...,m (\\textup{mod} m+1)$, and equals $m\\lceil\\frac{n}{s^\\star}\\rceil$ if $n\\equiv 0,1 (\\textup{mod} m+1)$, where $s^\\star$ is the minimum positive integer"},"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":"1208.0918","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.GR","submitted_at":"2012-08-04T12:28:01Z","cross_cats_sorted":[],"title_canon_sha256":"9976f229988bc85b054c0476212f383f0552687516678e58d4824fc06e897f28","abstract_canon_sha256":"d009104b75ca4fb9a5f8a5c1f540ca1e27e6eaa78593675c9f2ec2012ad8e7ef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:19:01.671174Z","signature_b64":"Y4yutYSFPMtRx29iV2ZTfu1lBllYdHFh87UqbVhOcp7pRrGGLdoXEadUNH0voOBuT+94BWoUkk4rmLxYEEY/CQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64cf380942ea44a7ff52feb83dd11eac5ee5ba2f78fb77a610f8ca60e0bc5396","last_reissued_at":"2026-05-18T03:19:01.670403Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:19:01.670403Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Equitable chromatic threshold of Kronecker products of complete graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.GR","authors_text":"Wei Wang, Zhidan Yan","submitted_at":"2012-08-04T12:28:01Z","abstract_excerpt":"A proper vertex coloring of a graph is equitable if the sizes of color classes differ by at most 1. The equitable chromatic threshold of a graph $G$, denoted by $\\chi_=^*(G)$, is the minimum $k$ such that $G$ is equitably $k^\\prime$-colorable for all $k^\\prime \\ge k$. Let $G\\times H$ denote the direct product of graphs $G$ and $H$. For $n\\ge m\\ge 2$ we prove that $\\chi_=^*(K_{m} \\times K_n)$ equals $\\lceil\\frac{mn}{m+1}\\rceil$ if $n\\equiv 2,...,m (\\textup{mod} m+1)$, and equals $m\\lceil\\frac{n}{s^\\star}\\rceil$ if $n\\equiv 0,1 (\\textup{mod} m+1)$, where $s^\\star$ is the minimum positive integer"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1208.0918","kind":"arxiv","version":3},"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":"1208.0918","created_at":"2026-05-18T03:19:01.670520+00:00"},{"alias_kind":"arxiv_version","alias_value":"1208.0918v3","created_at":"2026-05-18T03:19:01.670520+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1208.0918","created_at":"2026-05-18T03:19:01.670520+00:00"},{"alias_kind":"pith_short_12","alias_value":"MTHTQCKC5JCK","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_16","alias_value":"MTHTQCKC5JCKP72S","created_at":"2026-05-18T12:27:14.488303+00:00"},{"alias_kind":"pith_short_8","alias_value":"MTHTQCKC","created_at":"2026-05-18T12:27:14.488303+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/MTHTQCKC5JCKP72S724D3UI6VR","json":"https://pith.science/pith/MTHTQCKC5JCKP72S724D3UI6VR.json","graph_json":"https://pith.science/api/pith-number/MTHTQCKC5JCKP72S724D3UI6VR/graph.json","events_json":"https://pith.science/api/pith-number/MTHTQCKC5JCKP72S724D3UI6VR/events.json","paper":"https://pith.science/paper/MTHTQCKC"},"agent_actions":{"view_html":"https://pith.science/pith/MTHTQCKC5JCKP72S724D3UI6VR","download_json":"https://pith.science/pith/MTHTQCKC5JCKP72S724D3UI6VR.json","view_paper":"https://pith.science/paper/MTHTQCKC","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1208.0918&json=true","fetch_graph":"https://pith.science/api/pith-number/MTHTQCKC5JCKP72S724D3UI6VR/graph.json","fetch_events":"https://pith.science/api/pith-number/MTHTQCKC5JCKP72S724D3UI6VR/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/MTHTQCKC5JCKP72S724D3UI6VR/action/timestamp_anchor","attest_storage":"https://pith.science/pith/MTHTQCKC5JCKP72S724D3UI6VR/action/storage_attestation","attest_author":"https://pith.science/pith/MTHTQCKC5JCKP72S724D3UI6VR/action/author_attestation","sign_citation":"https://pith.science/pith/MTHTQCKC5JCKP72S724D3UI6VR/action/citation_signature","submit_replication":"https://pith.science/pith/MTHTQCKC5JCKP72S724D3UI6VR/action/replication_record"}},"created_at":"2026-05-18T03:19:01.670520+00:00","updated_at":"2026-05-18T03:19:01.670520+00:00"}