{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:OBGNIY6CYMH5NH66NAEDVLM2MB","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":"61b7c2539ab513f1f209a9fc3bbb45fc53c9a31aaf3f8c14b325ed1c6ab24d91","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-21T03:52:27Z","title_canon_sha256":"e0e97162ff96d8e34dfd387b624d498afdee92fd0d1caa01964c21ede46c1e24"},"schema_version":"1.0","source":{"id":"1504.05293","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1504.05293","created_at":"2026-05-18T02:18:14Z"},{"alias_kind":"arxiv_version","alias_value":"1504.05293v1","created_at":"2026-05-18T02:18:14Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1504.05293","created_at":"2026-05-18T02:18:14Z"},{"alias_kind":"pith_short_12","alias_value":"OBGNIY6CYMH5","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_16","alias_value":"OBGNIY6CYMH5NH66","created_at":"2026-05-18T12:29:34Z"},{"alias_kind":"pith_short_8","alias_value":"OBGNIY6C","created_at":"2026-05-18T12:29:34Z"}],"graph_snapshots":[{"event_id":"sha256:e05f146904b4de61fd33d783d3b03a763aeb4328d9ec7791053176c663c943d5","target":"graph","created_at":"2026-05-18T02:18:14Z","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":"Given 2-factors $R$ and $S$ of order $n$, let $r$ and $s$ be nonnegative integers with $r+s=\\lfloor \\frac{n-1}{2}\\rfloor$, the Hamilton-Waterloo problem asks for a 2-factorization of $K_n$ if $n$ is odd, or of $K_n-I$ if $n$ is even, in which $r$ of its 2-factors are isomorphic to $R$ and the other $s$ 2-factors are isomorphic to $S$. In this paper, we solve the problem for the case of triangle-factors and heptagon-factors for odd $n$ with 3 possible exceptions when $n=21$.","authors_text":"Hongchuan Lei, Hung-Lin Fu","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-21T03:52:27Z","title":"The Hamilton-Waterloo Problem for Triangle-Factors and Heptagon-Factors"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1504.05293","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":"sha256:90cdfc36f3bfa309ae5d1061c48378df6de37d5b97b65a4ae7366a376277898b","target":"record","created_at":"2026-05-18T02:18:14Z","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":"61b7c2539ab513f1f209a9fc3bbb45fc53c9a31aaf3f8c14b325ed1c6ab24d91","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2015-04-21T03:52:27Z","title_canon_sha256":"e0e97162ff96d8e34dfd387b624d498afdee92fd0d1caa01964c21ede46c1e24"},"schema_version":"1.0","source":{"id":"1504.05293","kind":"arxiv","version":1}},"canonical_sha256":"704cd463c2c30fd69fde68083aad9a60452235ead16806e68af97081e3e6f9fe","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"704cd463c2c30fd69fde68083aad9a60452235ead16806e68af97081e3e6f9fe","first_computed_at":"2026-05-18T02:18:14.627362Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T02:18:14.627362Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"njIafoIAlnVWILjQm2PTvZS16dOASS93YdRTxws7Ev8BdK88mrb83FHUTZDMJgpWd9X63+EZcv2ChQYm5OhSBQ==","signature_status":"signed_v1","signed_at":"2026-05-18T02:18:14.627956Z","signed_message":"canonical_sha256_bytes"},"source_id":"1504.05293","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:90cdfc36f3bfa309ae5d1061c48378df6de37d5b97b65a4ae7366a376277898b","sha256:e05f146904b4de61fd33d783d3b03a763aeb4328d9ec7791053176c663c943d5"],"state_sha256":"55a8f29bdd22717c92366e7c0409dff840b6b97f72cb54dff5d635b55b195f2a"}