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In this paper we prove that an $m$-colored semicomplete $r$-partite digraph $D$ has a $k$-colored kernel provided that $r\\geq 3$ and\n  {enumerate} [(i)] $k\\geq 4,$\n  [(ii)] $k=3$ and every $\\overrightarrow{C}_{4}$ contained in $D$ is at most 2-colored and, either every $\\overrightarrow{C"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1202.4017","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-17T20:56:11Z","cross_cats_sorted":[],"title_canon_sha256":"45ca6db480ae84f6dedda62f3053e89da27ed677e062a29ae496992126d4c83b","abstract_canon_sha256":"aaa2c5a474a05566e1bf01a3ba15b3e2dd1b4337677511cc21ff616027f5a32a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:02:02.175344Z","signature_b64":"0gGf5/IQlo0dVvBHmrBo4z9c4FRVOM1Dcoxe4Nv+DZFvATf484VJe5HeqWxoffy31Od5ADkZxYMPEaNc9nUHAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c8f572473bf626d913398c5ffad1c4743d0bfeb0bc8dcf349d7c165e9fb5a26e","last_reissued_at":"2026-05-18T04:02:02.174703Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:02:02.174703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$k$-colored kernels in semicomplete multipartite digraphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Bernardo Llano, Hortensia Galeana-S\\'anchez, Juan Jos\\'e Montellano-Ballesteros","submitted_at":"2012-02-17T20:56:11Z","abstract_excerpt":"An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K $ of its vertices such that for every vertex $v\\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and for every $% u,v\\in K$ there does not exist an at most $k$-colored directed path between them. In this paper we prove that an $m$-colored semicomplete $r$-partite digraph $D$ has a $k$-colored kernel provided that $r\\geq 3$ and\n  {enumerate} [(i)] $k\\geq 4,$\n  [(ii)] $k=3$ and every $\\overrightarrow{C}_{4}$ contained in $D$ is at most 2-colored and, either every $\\overrightarrow{C"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4017","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":"1202.4017","created_at":"2026-05-18T04:02:02.174818+00:00"},{"alias_kind":"arxiv_version","alias_value":"1202.4017v1","created_at":"2026-05-18T04:02:02.174818+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4017","created_at":"2026-05-18T04:02:02.174818+00:00"},{"alias_kind":"pith_short_12","alias_value":"ZD2XERZ36YTN","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_16","alias_value":"ZD2XERZ36YTNSEZZ","created_at":"2026-05-18T12:27:30.460161+00:00"},{"alias_kind":"pith_short_8","alias_value":"ZD2XERZ3","created_at":"2026-05-18T12:27:30.460161+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/ZD2XERZ36YTNSEZZRRP7VUOEOQ","json":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ.json","graph_json":"https://pith.science/api/pith-number/ZD2XERZ36YTNSEZZRRP7VUOEOQ/graph.json","events_json":"https://pith.science/api/pith-number/ZD2XERZ36YTNSEZZRRP7VUOEOQ/events.json","paper":"https://pith.science/paper/ZD2XERZ3"},"agent_actions":{"view_html":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ","download_json":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ.json","view_paper":"https://pith.science/paper/ZD2XERZ3","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1202.4017&json=true","fetch_graph":"https://pith.science/api/pith-number/ZD2XERZ36YTNSEZZRRP7VUOEOQ/graph.json","fetch_events":"https://pith.science/api/pith-number/ZD2XERZ36YTNSEZZRRP7VUOEOQ/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/action/timestamp_anchor","attest_storage":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/action/storage_attestation","attest_author":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/action/author_attestation","sign_citation":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/action/citation_signature","submit_replication":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/action/replication_record"}},"created_at":"2026-05-18T04:02:02.174818+00:00","updated_at":"2026-05-18T04:02:02.174818+00:00"}