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An $m$-colored digraph $D$ has $k$-colored kernel if there exists a subset $K$ of its vertices such that\n  (i) from every vertex $v\\notin K$ there exists an at most $k$-colored directed path from $v$ to a vertex of $K$ and\n  (ii) for every $u,v\\in K$ there does not exist an at most $k$-colored directed path between them.\n  In this paper, we prove that for every integer $k\\geq 2$ there exists a $% (k+1)$-colored digraph $D$ without $k$-colored kernel and if every directed cycle of an $m$-colored digraph is monochromatic, then it has a $k$-co"},"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":"1201.2712","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-01-13T00:18:15Z","cross_cats_sorted":[],"title_canon_sha256":"552a63c8be7326a027ebbfcb09676a98ef775f43e154881ac2db1f398da0421d","abstract_canon_sha256":"3fd0173d409939429c61b5a28cc28fe74ecd23228e02ad414b1942ab632bbc62"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:04:43.259229Z","signature_b64":"9pe+Pu6FXBlvB83RToV66SnyZ55RgMLIgEb5sf39Kk14R/yXMZWySGvuxsyq6AjIkQyEDDKhmr+JDv56H9bBAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2f3b223c2133479b82288c25dfd590d83f60512e94a469d84e115b10f43d1e8a","last_reissued_at":"2026-05-18T04:04:43.258748Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:04:43.258748Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"k-colored kernels","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":"2012-01-13T00:18:15Z","abstract_excerpt":"We study $k$-colored kernels in $m$-colored digraphs. 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