{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:7IWQ5SCM53CFBJZEWPXTCW5GVD","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":"53851c240b485f5367d0285abcd8f1f10fd779d5dc97b8339433d008147bc67f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-30T11:53:08Z","title_canon_sha256":"018c3f11ebaaff529adec9a811360f4aeba61235b4b0cdea479219b51a687592"},"schema_version":"1.0","source":{"id":"1608.00105","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1608.00105","created_at":"2026-05-18T00:55:51Z"},{"alias_kind":"arxiv_version","alias_value":"1608.00105v3","created_at":"2026-05-18T00:55:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1608.00105","created_at":"2026-05-18T00:55:51Z"},{"alias_kind":"pith_short_12","alias_value":"7IWQ5SCM53CF","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_16","alias_value":"7IWQ5SCM53CFBJZE","created_at":"2026-05-18T12:30:04Z"},{"alias_kind":"pith_short_8","alias_value":"7IWQ5SCM","created_at":"2026-05-18T12:30:04Z"}],"graph_snapshots":[{"event_id":"sha256:a72af148cc5e4c529d9d47055858081c741691f970d3b1c15fe0816f58b3abeb","target":"graph","created_at":"2026-05-18T00:55:51Z","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":"Let $G$ be a nontrivial connected graph of order $n$ with an edge-coloring $c:E(G)\\rightarrow\\{1,2,\\dots,t\\}$,$t\\in\\mathbb{N}$, where adjacent edges may be colored with the same color. A tree $T$ in $G$ is a \\emph{proper tree} if no two adjacent edges of it are assigned the same color. Let $k$ be a fixed integer with $2\\leq k\\leq n$. For a vertex subset $S\\subseteq V(G)$ with $|S|\\geq 2$, a tree is called an \\emph{$S$-tree} if it connects $S$ in $G$ . A \\emph{$k$-proper coloring} of $G$ is an edge-coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a","authors_text":"Jingshu Zhang, Wenjing Li, Xueliang Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-30T11:53:08Z","title":"The $k$-proper index of complete bipartite and complete multipartite graphs"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.00105","kind":"arxiv","version":3},"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:e876eb14b4640195f178ed3973cc4318e05e339f394a62c1508dbb9626df92f7","target":"record","created_at":"2026-05-18T00:55:51Z","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":"53851c240b485f5367d0285abcd8f1f10fd779d5dc97b8339433d008147bc67f","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2016-07-30T11:53:08Z","title_canon_sha256":"018c3f11ebaaff529adec9a811360f4aeba61235b4b0cdea479219b51a687592"},"schema_version":"1.0","source":{"id":"1608.00105","kind":"arxiv","version":3}},"canonical_sha256":"fa2d0ec84ceec450a724b3ef315ba6a8de9b404cc5f2e8d44e97d5e1f69af4c0","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"fa2d0ec84ceec450a724b3ef315ba6a8de9b404cc5f2e8d44e97d5e1f69af4c0","first_computed_at":"2026-05-18T00:55:51.836188Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:55:51.836188Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zJUvGYk6NtEGxUjTUo52DBPqXVYGEZYliX/BMkqfW0A53fAv3Ko6AdWuDOkvwb6uFfjDJxOZgrbbqPtSX7fwAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:55:51.836682Z","signed_message":"canonical_sha256_bytes"},"source_id":"1608.00105","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e876eb14b4640195f178ed3973cc4318e05e339f394a62c1508dbb9626df92f7","sha256:a72af148cc5e4c529d9d47055858081c741691f970d3b1c15fe0816f58b3abeb"],"state_sha256":"f57991b0519267fe66c89b7b2db5d2b8191e901771e81bdb52f6e9f0509ee003"}