{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2017:7A2Z7LKTYYI3SU2VGNTZITVTVH","short_pith_number":"pith:7A2Z7LKT","schema_version":"1.0","canonical_sha256":"f8359fad53c611b953553367944eb3a9f7a3127a2d99af2c38813dcb8ba068ad","source":{"kind":"arxiv","id":"1712.09291","version":1},"attestation_state":"computed","paper":{"title":"On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adri\\'an Pastine, Melissa Keranen","submitted_at":"2017-12-26T15:47:57Z","abstract_excerpt":"The Hamilton-Waterloo problem asks for a decomposition of the complete graph into $r$ copies of a 2-factor $F_{1}$ and $s$ copies of a 2-factor $F_{2}$ such that $r+s=\\left\\lfloor\\frac{v-1}{2}\\right\\rfloor$. If $F_{1}$ consists of $m$-cycles and $F_{2}$ consists of $n$ cycles, then we call such a decomposition a $(m,n)-$HWP$(v;r,s)$. The goal is to find a decomposition for every possible pair $(r,s)$. In this paper, we show that for odd $x$ and $y$, there is a $(2^kx,y)-$HWP$(vm;r,s)$ if $\\gcd(x,y)\\geq 3$, $m\\geq 3$, and both $x$ and $y$ divide $v$, except possibly when $1\\in\\{r,s\\}$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1712.09291","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-12-26T15:47:57Z","cross_cats_sorted":[],"title_canon_sha256":"b7b267e50d1a19defbbab791b6539200904a56ad09de2ca3e4a4c83c02e12fda","abstract_canon_sha256":"ab826825853d9315f34b8603b2b97dfa6dc70318bb35880b33c508c6fe626149"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:27:14.016279Z","signature_b64":"GKcos9K0HWOjh/mitPeVgpUGbw+7wMrUTli+QUFsedw/XGO4U8Jg+2KGgSjc11F0ofrRabZ5kROs+rdE4+5bCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8359fad53c611b953553367944eb3a9f7a3127a2d99af2c38813dcb8ba068ad","last_reissued_at":"2026-05-18T00:27:14.015854Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:27:14.015854Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Hamilton-Waterloo problem: the case of two cycles sizes of different parity","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adri\\'an Pastine, Melissa Keranen","submitted_at":"2017-12-26T15:47:57Z","abstract_excerpt":"The Hamilton-Waterloo problem asks for a decomposition of the complete graph into $r$ copies of a 2-factor $F_{1}$ and $s$ copies of a 2-factor $F_{2}$ such that $r+s=\\left\\lfloor\\frac{v-1}{2}\\right\\rfloor$. If $F_{1}$ consists of $m$-cycles and $F_{2}$ consists of $n$ cycles, then we call such a decomposition a $(m,n)-$HWP$(v;r,s)$. The goal is to find a decomposition for every possible pair $(r,s)$. In this paper, we show that for odd $x$ and $y$, there is a $(2^kx,y)-$HWP$(vm;r,s)$ if $\\gcd(x,y)\\geq 3$, $m\\geq 3$, and both $x$ and $y$ divide $v$, except possibly when $1\\in\\{r,s\\}$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1712.09291","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1712.09291","created_at":"2026-05-18T00:27:14.015914+00:00"},{"alias_kind":"arxiv_version","alias_value":"1712.09291v1","created_at":"2026-05-18T00:27:14.015914+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1712.09291","created_at":"2026-05-18T00:27:14.015914+00:00"},{"alias_kind":"pith_short_12","alias_value":"7A2Z7LKTYYI3","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_16","alias_value":"7A2Z7LKTYYI3SU2V","created_at":"2026-05-18T12:31:03.183658+00:00"},{"alias_kind":"pith_short_8","alias_value":"7A2Z7LKT","created_at":"2026-05-18T12:31:03.183658+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH","json":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH.json","graph_json":"https://pith.science/api/pith-number/7A2Z7LKTYYI3SU2VGNTZITVTVH/graph.json","events_json":"https://pith.science/api/pith-number/7A2Z7LKTYYI3SU2VGNTZITVTVH/events.json","paper":"https://pith.science/paper/7A2Z7LKT"},"agent_actions":{"view_html":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH","download_json":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH.json","view_paper":"https://pith.science/paper/7A2Z7LKT","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1712.09291&json=true","fetch_graph":"https://pith.science/api/pith-number/7A2Z7LKTYYI3SU2VGNTZITVTVH/graph.json","fetch_events":"https://pith.science/api/pith-number/7A2Z7LKTYYI3SU2VGNTZITVTVH/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH/action/storage_attestation","attest_author":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH/action/author_attestation","sign_citation":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH/action/citation_signature","submit_replication":"https://pith.science/pith/7A2Z7LKTYYI3SU2VGNTZITVTVH/action/replication_record"}},"created_at":"2026-05-18T00:27:14.015914+00:00","updated_at":"2026-05-18T00:27:14.015914+00:00"}