{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:L5AYSP3UFIJMTUS6ZNNFYP3NGI","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":"15f5931421aac95c35b43d7931b5c66799392295c87829ed575fc1debe9ece72","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-20T13:35:02Z","title_canon_sha256":"c1bd97c25c5b4d7dc2a49528641e8192598054de0eb967998869c06ad8231007"},"schema_version":"1.0","source":{"id":"1301.4657","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1301.4657","created_at":"2026-05-18T03:35:25Z"},{"alias_kind":"arxiv_version","alias_value":"1301.4657v2","created_at":"2026-05-18T03:35:25Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.4657","created_at":"2026-05-18T03:35:25Z"},{"alias_kind":"pith_short_12","alias_value":"L5AYSP3UFIJM","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_16","alias_value":"L5AYSP3UFIJMTUS6","created_at":"2026-05-18T12:27:51Z"},{"alias_kind":"pith_short_8","alias_value":"L5AYSP3U","created_at":"2026-05-18T12:27:51Z"}],"graph_snapshots":[{"event_id":"sha256:069b142c64f497a0cbb2ebbcc95e603b71c0bc1c2f5e1040a8136008ba21d69a","target":"graph","created_at":"2026-05-18T03:35:25Z","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 graph with vertex set $V(G)$ and let $H:V(G)\\rightarrow 2^N$ be a set function associating with $G$. An $H$-factor of graph $G$ is a spanning subgraphs $F$ such that $$d_F(v)\\in H(v){4em}\\hbox{for every}v\\in V(G).$$ Let $f:V(G)\\rightarrow N$ be an even integer-valued function such that $f\\geq 4$ and let $H_f(v)=\\{1,3,...,f(v)-1, f(v)\\}$ for $v\\in V(G)$. In this paper, we investigate $H_f$-factors of graphs $G$ by using Lov\\'asz's structural descriptions. Let $o(G)$ denote the number of odd components of $G$.\n  We show that if one of the following conditions holds, then $G$ contain","authors_text":"Hongliang Lu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-20T13:35:02Z","title":"An Extension of Cui-Kano's Characterization Problem on Graph Factors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.4657","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:b50451deadd16336d5651b302cb4d09e4cf58f150c5da1c1e1f59e0d978248e0","target":"record","created_at":"2026-05-18T03:35:25Z","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":"15f5931421aac95c35b43d7931b5c66799392295c87829ed575fc1debe9ece72","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2013-01-20T13:35:02Z","title_canon_sha256":"c1bd97c25c5b4d7dc2a49528641e8192598054de0eb967998869c06ad8231007"},"schema_version":"1.0","source":{"id":"1301.4657","kind":"arxiv","version":2}},"canonical_sha256":"5f41893f742a12c9d25ecb5a5c3f6d3210c3c58a59270ea10e6a08cf270a6ff5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"5f41893f742a12c9d25ecb5a5c3f6d3210c3c58a59270ea10e6a08cf270a6ff5","first_computed_at":"2026-05-18T03:35:25.029904Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:35:25.029904Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"CoQbMqJ+xdrKm5RPUBC0GnMiv4YgQqBbUgtGSiw81AbHO+ux9PiL2BIqwypWlV1cHrSU6ZyYH/CkgzSAIudUDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:35:25.030576Z","signed_message":"canonical_sha256_bytes"},"source_id":"1301.4657","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b50451deadd16336d5651b302cb4d09e4cf58f150c5da1c1e1f59e0d978248e0","sha256:069b142c64f497a0cbb2ebbcc95e603b71c0bc1c2f5e1040a8136008ba21d69a"],"state_sha256":"7ef67c2caae904ecb7b5e712d2877629142f14170364da4484308905647a6176"}