{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:2Y3JXFMYXCA3AKYLRX6E6IDFNX","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":"6a01a08e05160878ca07eaafcaca72dc3dac588cf481c4a723eabd8ec4b86a49","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-10T07:31:54Z","title_canon_sha256":"f6b86243db185c5c6cdfa9af0343cf97550432a5d3c70bbb38d8cc2be49ad0f2"},"schema_version":"1.0","source":{"id":"2606.11757","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.11757","created_at":"2026-06-11T01:10:06Z"},{"alias_kind":"arxiv_version","alias_value":"2606.11757v1","created_at":"2026-06-11T01:10:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.11757","created_at":"2026-06-11T01:10:06Z"},{"alias_kind":"pith_short_12","alias_value":"2Y3JXFMYXCA3","created_at":"2026-06-11T01:10:06Z"},{"alias_kind":"pith_short_16","alias_value":"2Y3JXFMYXCA3AKYL","created_at":"2026-06-11T01:10:06Z"},{"alias_kind":"pith_short_8","alias_value":"2Y3JXFMY","created_at":"2026-06-11T01:10:06Z"}],"graph_snapshots":[{"event_id":"sha256:3b9b8c54734af1e7c572ee28bbeb9aaf4bda74dc8d7ae03c7fc8c4b2202bcdf7","target":"graph","created_at":"2026-06-11T01:10:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.11757/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"A graph $G$ is $k$-$colorable$ if $V(G)$ can be partitioned into at most $k$ stable sets. A graph $G$ is $k$-$chromatic$ if $k$ is the smallest integer for which $G$ is $k$-colorable. In general, for a fixed $k\\ge 3$, determining whether an arbitrary graph $G$ is $k$-colorable is NP-complete. Consequently, $k$-coloring algorithms for restricted graph classes, such as $\\mathcal{H}$-free graphs, have been widely studied over the past few decades. A graph $G$ is $k$-$vertex$-$critical$ if $G$ is $k$-chromatic and every proper induced subgraph of $G$ is ($k$-1)-colorable. Given a graph $G$, most o","authors_text":"Manoj Belavadi, T. Karthick","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-10T07:31:54Z","title":"Vertex-critical co-gem-free graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.11757","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":"sha256:358f84d0bf44202ca423203b27ba40236c3e47f47fe2ad7da13fcdfa3bd01136","target":"record","created_at":"2026-06-11T01:10:06Z","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":"6a01a08e05160878ca07eaafcaca72dc3dac588cf481c4a723eabd8ec4b86a49","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","primary_cat":"math.CO","submitted_at":"2026-06-10T07:31:54Z","title_canon_sha256":"f6b86243db185c5c6cdfa9af0343cf97550432a5d3c70bbb38d8cc2be49ad0f2"},"schema_version":"1.0","source":{"id":"2606.11757","kind":"arxiv","version":1}},"canonical_sha256":"d6369b9598b881b02b0b8dfc4f20656dcf85328755d021686eb595fd35c187ef","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"d6369b9598b881b02b0b8dfc4f20656dcf85328755d021686eb595fd35c187ef","first_computed_at":"2026-06-11T01:10:06.389536Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-11T01:10:06.389536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"jNSlQVOdEcH1GDbUpmRVjZvlf+P/Q51p09rH8VLNdzfGFdqmaxIZaVlSK8XXDqz9r6IrTxjsJEYjOJ9BcQ0nCw==","signature_status":"signed_v1","signed_at":"2026-06-11T01:10:06.390328Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.11757","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:358f84d0bf44202ca423203b27ba40236c3e47f47fe2ad7da13fcdfa3bd01136","sha256:3b9b8c54734af1e7c572ee28bbeb9aaf4bda74dc8d7ae03c7fc8c4b2202bcdf7"],"state_sha256":"2953262071a36137e396b3642ac8afaf5f6b10300e6f8a87d97c27c6efa5309a"}