{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:KOGVDNNL32TVOB5YXD7A6WCAZ3","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":"f84168b856714dad0a38a5d8021d3cf9579864daa6c702ddeca354eed162710f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-02T14:52:50Z","title_canon_sha256":"622f600db9e3eac09e321cb12257a58c207b5f871d41d3752da1f9ff251cad6d"},"schema_version":"1.0","source":{"id":"2607.02270","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2607.02270","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"arxiv_version","alias_value":"2607.02270v1","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2607.02270","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"pith_short_12","alias_value":"KOGVDNNL32TV","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"pith_short_16","alias_value":"KOGVDNNL32TVOB5Y","created_at":"2026-07-03T01:17:47Z"},{"alias_kind":"pith_short_8","alias_value":"KOGVDNNL","created_at":"2026-07-03T01:17:47Z"}],"graph_snapshots":[{"event_id":"sha256:86781ba2675be05fd1b1ef7b1a6da8daf5590d77eb857a3341b80338c91660d3","target":"graph","created_at":"2026-07-03T01:17:47Z","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/2607.02270/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Let $G$ be a simple graph with maximum degree $\\Delta(G)$. A subgraph $H\\subseteq G$ is $\\Delta(G)$-overfull if $|E(H)|>\\Delta(G)\\left\\lfloor |V(H)|/2\\right\\rfloor$. In any edge coloring of $G$, each color class restricted to $H$ is a matching of size at most $\\left\\lfloor |V(H)|/2\\right\\rfloor$. Thus, if $G$ contains a $\\Delta(G)$-overfull subgraph, then $G$ cannot be edge-colored with only $\\Delta(G)$ colors. By Vizing's Theorem, $\\chi'(G)\\le \\Delta(G)+1$, and hence $G$ is class $2$. In 1986, Chetwynd and Hilton conjectured that whenever $\\Delta(G)>|V(G)|/3$, the converse also holds: every c","authors_text":"Guantao Chen, Jessica McDonald, Songling Shan","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-02T14:52:50Z","title":"Towards the Overfull Conjecture II"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2607.02270","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:dcfc68c566961e3b55d09bf096c34cbe2f7bca315a7101b356206be380ae8a81","target":"record","created_at":"2026-07-03T01:17:47Z","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":"f84168b856714dad0a38a5d8021d3cf9579864daa6c702ddeca354eed162710f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2026-07-02T14:52:50Z","title_canon_sha256":"622f600db9e3eac09e321cb12257a58c207b5f871d41d3752da1f9ff251cad6d"},"schema_version":"1.0","source":{"id":"2607.02270","kind":"arxiv","version":1}},"canonical_sha256":"538d51b5abdea75707b8b8fe0f5840cefe495bcdbe578dbde663dbaa355debc7","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"538d51b5abdea75707b8b8fe0f5840cefe495bcdbe578dbde663dbaa355debc7","first_computed_at":"2026-07-03T01:17:47.120259Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-03T01:17:47.120259Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"b38c/zFxpIpebJO/6JvVS/5LU7bB6s44GaB3/sRpJqBSQ3LMRn2k0Of2Xo6Mykpf9Z1aSAVByF0uiYcoF/PRBw==","signature_status":"signed_v1","signed_at":"2026-07-03T01:17:47.120711Z","signed_message":"canonical_sha256_bytes"},"source_id":"2607.02270","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:dcfc68c566961e3b55d09bf096c34cbe2f7bca315a7101b356206be380ae8a81","sha256:86781ba2675be05fd1b1ef7b1a6da8daf5590d77eb857a3341b80338c91660d3"],"state_sha256":"59f867af4f068fe1c4c1151e81ef2bc8d973ab1a6aacab9e422b59b62d5bb3c0"}