{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2014:LN4UB6JFEFJXOR3GF6ZD5PW2EB","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":"2abb12b7b2451217254698b4667f383232983d408ded4020f6e7592737577a31","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-03-31T14:47:40Z","title_canon_sha256":"d3ca4b9993bb58480c437327741edc8550e77ffe547f179fd15c138b07c10889"},"schema_version":"1.0","source":{"id":"1403.8027","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1403.8027","created_at":"2026-05-18T02:32:38Z"},{"alias_kind":"arxiv_version","alias_value":"1403.8027v2","created_at":"2026-05-18T02:32:38Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1403.8027","created_at":"2026-05-18T02:32:38Z"},{"alias_kind":"pith_short_12","alias_value":"LN4UB6JFEFJX","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_16","alias_value":"LN4UB6JFEFJXOR3G","created_at":"2026-05-18T12:28:38Z"},{"alias_kind":"pith_short_8","alias_value":"LN4UB6JF","created_at":"2026-05-18T12:28:38Z"}],"graph_snapshots":[{"event_id":"sha256:b17d35ab34caf6eb044372785289a7f26579bf507c1ed3ac280618ae81e9e595","target":"graph","created_at":"2026-05-18T02:32:38Z","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":"A graph is $k$-critical if it is $k$-chromatic but each of its proper induced subgraphs is ($k-1$)-colorable. It is known that the number of $4$-critical $P_5$-free graphs is finite, but there is an infinite number of $k$-critical $P_5$-free graphs for each $k \\geq 5$. We show that the number of $k$-critical $(P_5, \\overline{P}_5)$-free graphs is finite for every fixed $k$. Our result implies the existence of a certifying algorithm for $k$-coloring $(P_5, \\overline{P}_5)$-free graphs.","authors_text":"Ang\\`ele M. Hamel, Ch\\'inh T. Ho\\`ang, Fr\\'ed\\'eric Maffray, Harjinder S. Dhaliwal, Stefan A. Panait, Tyler J. D. McConnell","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-03-31T14:47:40Z","title":"On color-critical ($P_{5},\\overline{P}_5$)-free graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1403.8027","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:a87ba977962175c41d37917ccac9992b64245c98d18c4e975633b7350ca58d2a","target":"record","created_at":"2026-05-18T02:32:38Z","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":"2abb12b7b2451217254698b4667f383232983d408ded4020f6e7592737577a31","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2014-03-31T14:47:40Z","title_canon_sha256":"d3ca4b9993bb58480c437327741edc8550e77ffe547f179fd15c138b07c10889"},"schema_version":"1.0","source":{"id":"1403.8027","kind":"arxiv","version":2}},"canonical_sha256":"5b7940f92521537747662fb23ebeda204f3e20eebf17a457cc06ec84f07c0f07","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5b7940f92521537747662fb23ebeda204f3e20eebf17a457cc06ec84f07c0f07","first_computed_at":"2026-05-18T02:32:38.853939Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:32:38.853939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"EM/l6IDIJ7/UEXciBixBu5ekLp4iV5gaBHEom9l1HyWDer4XFHtW8bqNGxtqfCcsov4mfc99UjUQFpHhfRSbBw==","signature_status":"signed_v1","signed_at":"2026-05-18T02:32:38.854282Z","signed_message":"canonical_sha256_bytes"},"source_id":"1403.8027","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:a87ba977962175c41d37917ccac9992b64245c98d18c4e975633b7350ca58d2a","sha256:b17d35ab34caf6eb044372785289a7f26579bf507c1ed3ac280618ae81e9e595"],"state_sha256":"6c453135d75a53313e42fe2c121f705b1a5918361aa3e27f0e5e92673d069d6a"}