{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:QHGQ4AZGG2B6VP45HGWCISLSZZ","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":"741650801ae25120e91d81a9c9b42fb53f43a5dc44cf2ab45859ee7a48815202","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-13T18:55:26Z","title_canon_sha256":"8908238c1af585ab6f688002d339710b2fe99ac01ce357d7a640d98b33fa9f00"},"schema_version":"1.0","source":{"id":"1701.03780","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1701.03780","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"arxiv_version","alias_value":"1701.03780v2","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1701.03780","created_at":"2026-05-18T00:20:22Z"},{"alias_kind":"pith_short_12","alias_value":"QHGQ4AZGG2B6","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_16","alias_value":"QHGQ4AZGG2B6VP45","created_at":"2026-05-18T12:31:39Z"},{"alias_kind":"pith_short_8","alias_value":"QHGQ4AZG","created_at":"2026-05-18T12:31:39Z"}],"graph_snapshots":[{"event_id":"sha256:dd00c142d86f958761d7cf3821039e9092d170b9a98cd0d4818fe8982800fcaa","target":"graph","created_at":"2026-05-18T00:20:22Z","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":"The purpose of this note is to draw attention to problems related to a concept called majority colouring recently studied by Kreutzer, Oum, Seymour, van der Zypen and Wood. They raised a problem of determining, for a natural number $k$, the smallest number $m=m(k)$ such that every digraph can be coloured with $m$ colours where each vertex has the same colour as at most $1/k$ proportion of its out-neighbours. We show that $m(k)\\in\\{2k-1,2k\\}$. We also prove a result supporting the conjecture that $m(2)=3$. Moreover, we prove similar results for a more general concept called majority choosabilit","authors_text":"Ant\\'onio Gir\\~ao, Kamil Popielarz, Teeradej Kittipassorn","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-13T18:55:26Z","title":"Generalised Majority Colourings of Digraphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1701.03780","kind":"arxiv","version":2},"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":"sha256:0e23963b8df45573cf2efed9227276c2f0ef8a2d7df82536c234b5fd26ae9b98","target":"record","created_at":"2026-05-18T00:20:22Z","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":"741650801ae25120e91d81a9c9b42fb53f43a5dc44cf2ab45859ee7a48815202","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-01-13T18:55:26Z","title_canon_sha256":"8908238c1af585ab6f688002d339710b2fe99ac01ce357d7a640d98b33fa9f00"},"schema_version":"1.0","source":{"id":"1701.03780","kind":"arxiv","version":2}},"canonical_sha256":"81cd0e03263683eabf9d39ac244972ce6f4141454753d80cf06bfac0eecfa3f3","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81cd0e03263683eabf9d39ac244972ce6f4141454753d80cf06bfac0eecfa3f3","first_computed_at":"2026-05-18T00:20:22.379770Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:20:22.379770Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"YpjCYeO7MoS8aCsDriJy8vHmxVlKerL539nrClBO6bLq9UWCSc0p0hNl0gJqaBbmz73GFlhxzsnt6FkFRygFDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:20:22.380352Z","signed_message":"canonical_sha256_bytes"},"source_id":"1701.03780","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:0e23963b8df45573cf2efed9227276c2f0ef8a2d7df82536c234b5fd26ae9b98","sha256:dd00c142d86f958761d7cf3821039e9092d170b9a98cd0d4818fe8982800fcaa"],"state_sha256":"eff4fd176c232a5277889aa2042c30a435bcc59040e99da4e71615ecc38d2c22"}