{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:FLE2Q64LISCASBZUZHNYYPNZTZ","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":"1d117bc56362e7163138ebd5373c16309a0ffae470ab5b8fccf130f38bfd5284","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-23T19:08:08Z","title_canon_sha256":"93134465003cd0dbdff9b3515641e5936c830cc21c00973e3d24b10f93b8580b"},"schema_version":"1.0","source":{"id":"1510.07017","kind":"arxiv","version":4}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1510.07017","created_at":"2026-05-18T00:44:37Z"},{"alias_kind":"arxiv_version","alias_value":"1510.07017v4","created_at":"2026-05-18T00:44:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1510.07017","created_at":"2026-05-18T00:44:37Z"},{"alias_kind":"pith_short_12","alias_value":"FLE2Q64LISCA","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_16","alias_value":"FLE2Q64LISCASBZU","created_at":"2026-05-18T12:29:19Z"},{"alias_kind":"pith_short_8","alias_value":"FLE2Q64L","created_at":"2026-05-18T12:29:19Z"}],"graph_snapshots":[{"event_id":"sha256:ce30a244731ee79493da8c99891beb1aff79701ce16f36512d87351e4d8f9ad0","target":"graph","created_at":"2026-05-18T00:44:37Z","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":"We prove that if $M$ is a maximal $k$-edge-colorable subgraph of a multigraph $G$ and if $F = \\{v \\in V(G) : d_M(v) \\leq k-\\mu(v)\\}$, then $d_F(v) \\leq d_M(v)$ for all $v \\in F$. (When $G$ is a simple graph, the set $F$ is just the set of vertices having degree less than $k$ in $M$.) This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.","authors_text":"Gregory J. Puleo","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-23T19:08:08Z","title":"Maximal $k$-Edge-Colorable Subgraphs, Vizing's Theorem, and Tuza's Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.07017","kind":"arxiv","version":4},"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":"sha256:2c540160fe2e0574b0bb674075aff3567443bc36234faf16c935e966287f6393","target":"record","created_at":"2026-05-18T00:44:37Z","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":"1d117bc56362e7163138ebd5373c16309a0ffae470ab5b8fccf130f38bfd5284","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-10-23T19:08:08Z","title_canon_sha256":"93134465003cd0dbdff9b3515641e5936c830cc21c00973e3d24b10f93b8580b"},"schema_version":"1.0","source":{"id":"1510.07017","kind":"arxiv","version":4}},"canonical_sha256":"2ac9a87b8b4484090734c9db8c3db99e4a2972a80103471b740f505a4734248b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2ac9a87b8b4484090734c9db8c3db99e4a2972a80103471b740f505a4734248b","first_computed_at":"2026-05-18T00:44:37.264084Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:44:37.264084Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"SqyF6E30o2Tnc4qynLwOYmreTGTs2C1ts3CpebHxuZALA/MkB6E/iAsmZ1FtBbgXs2wxPsnPUvI2lalycpR1AQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:44:37.264629Z","signed_message":"canonical_sha256_bytes"},"source_id":"1510.07017","source_kind":"arxiv","source_version":4}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2c540160fe2e0574b0bb674075aff3567443bc36234faf16c935e966287f6393","sha256:ce30a244731ee79493da8c99891beb1aff79701ce16f36512d87351e4d8f9ad0"],"state_sha256":"db3c31274e1b5a8d8636767a00410c0c7d6ba3ebc67558c372d1c0f3ecfada75"}