{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2012:ZD2XERZ36YTNSEZZRRP7VUOEOQ","short_pith_number":"pith:ZD2XERZ3","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"},"canonical_sha256":"c8f572473bf626d913398c5ffad1c4743d0bfeb0bc8dcf349d7c165e9fb5a26e","source":{"kind":"arxiv","id":"1202.4017","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.4017","created_at":"2026-05-18T04:02:02Z"},{"alias_kind":"arxiv_version","alias_value":"1202.4017v1","created_at":"2026-05-18T04:02:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4017","created_at":"2026-05-18T04:02:02Z"},{"alias_kind":"pith_short_12","alias_value":"ZD2XERZ36YTN","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZD2XERZ36YTNSEZZ","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZD2XERZ3","created_at":"2026-05-18T12:27:30Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2012:ZD2XERZ36YTNSEZZRRP7VUOEOQ","target":"record","payload":{"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"},"canonical_sha256":"c8f572473bf626d913398c5ffad1c4743d0bfeb0bc8dcf349d7c165e9fb5a26e","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"},"source_kind":"arxiv","source_id":"1202.4017","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:02:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"APo3/VNF8MbQQh+F5zrYa8rpXkUf7T4HqLpgrmdi7voKjkMHNU+AlDZcS4iXrgTOL37+JedMmEBm4Uy6R4W0Bg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T06:51:01.888492Z"},"content_sha256":"2fa9bd7bb9f010c8dfee916d57484012b0605a323d64b67f41bfcf1f03f7f05b","schema_version":"1.0","event_id":"sha256:2fa9bd7bb9f010c8dfee916d57484012b0605a323d64b67f41bfcf1f03f7f05b"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2012:ZD2XERZ36YTNSEZZRRP7VUOEOQ","target":"graph","payload":{"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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:02:02Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mJ1J7XBAtGm/109TU9x/MFPWBreAjYxmcdhhXcEW9IHnK+2BsTZSh67pub9gS6Or6L/yGNduvtFkALRtGjGdBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-04T06:51:01.888846Z"},"content_sha256":"30754772525610ff4a61f051258148544564461dc48cf0879907bbc32c4c6c55","schema_version":"1.0","event_id":"sha256:30754772525610ff4a61f051258148544564461dc48cf0879907bbc32c4c6c55"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/bundle.json","state_url":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-04T06:51:01Z","links":{"resolver":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ","bundle":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/bundle.json","state":"https://pith.science/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/ZD2XERZ36YTNSEZZRRP7VUOEOQ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2012:ZD2XERZ36YTNSEZZRRP7VUOEOQ","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"aaa2c5a474a05566e1bf01a3ba15b3e2dd1b4337677511cc21ff616027f5a32a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-17T20:56:11Z","title_canon_sha256":"45ca6db480ae84f6dedda62f3053e89da27ed677e062a29ae496992126d4c83b"},"schema_version":"1.0","source":{"id":"1202.4017","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1202.4017","created_at":"2026-05-18T04:02:02Z"},{"alias_kind":"arxiv_version","alias_value":"1202.4017v1","created_at":"2026-05-18T04:02:02Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1202.4017","created_at":"2026-05-18T04:02:02Z"},{"alias_kind":"pith_short_12","alias_value":"ZD2XERZ36YTN","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_16","alias_value":"ZD2XERZ36YTNSEZZ","created_at":"2026-05-18T12:27:30Z"},{"alias_kind":"pith_short_8","alias_value":"ZD2XERZ3","created_at":"2026-05-18T12:27:30Z"}],"graph_snapshots":[{"event_id":"sha256:30754772525610ff4a61f051258148544564461dc48cf0879907bbc32c4c6c55","target":"graph","created_at":"2026-05-18T04:02:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"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 $\\overrightarrow{C","authors_text":"Bernardo Llano, Hortensia Galeana-S\\'anchez, Juan Jos\\'e Montellano-Ballesteros","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-17T20:56:11Z","title":"$k$-colored kernels in semicomplete multipartite digraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1202.4017","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:2fa9bd7bb9f010c8dfee916d57484012b0605a323d64b67f41bfcf1f03f7f05b","target":"record","created_at":"2026-05-18T04:02:02Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"aaa2c5a474a05566e1bf01a3ba15b3e2dd1b4337677511cc21ff616027f5a32a","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2012-02-17T20:56:11Z","title_canon_sha256":"45ca6db480ae84f6dedda62f3053e89da27ed677e062a29ae496992126d4c83b"},"schema_version":"1.0","source":{"id":"1202.4017","kind":"arxiv","version":1}},"canonical_sha256":"c8f572473bf626d913398c5ffad1c4743d0bfeb0bc8dcf349d7c165e9fb5a26e","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"c8f572473bf626d913398c5ffad1c4743d0bfeb0bc8dcf349d7c165e9fb5a26e","first_computed_at":"2026-05-18T04:02:02.174703Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:02:02.174703Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0gGf5/IQlo0dVvBHmrBo4z9c4FRVOM1Dcoxe4Nv+DZFvATf484VJe5HeqWxoffy31Od5ADkZxYMPEaNc9nUHAg==","signature_status":"signed_v1","signed_at":"2026-05-18T04:02:02.175344Z","signed_message":"canonical_sha256_bytes"},"source_id":"1202.4017","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2fa9bd7bb9f010c8dfee916d57484012b0605a323d64b67f41bfcf1f03f7f05b","sha256:30754772525610ff4a61f051258148544564461dc48cf0879907bbc32c4c6c55"],"state_sha256":"00a1b64dcad04e14756e611a1681fa8e5f6a4a50a1670b35b2468924b320c1b7"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"NxlPNq0GEwKlRytIomyJpr9EqVFTxRrfPvKiS4F3DPvEFVu0W/cSkndANbCcBYMy0BdXKATpA9jX+pwY5KYwDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-04T06:51:01.890982Z","bundle_sha256":"b807b3da3899b219a5cec7b182e18e5c51588a08e9d2ae7d80560115b4fce17d"}}