{"paper":{"title":"On the Hamilton-Waterloo Problem with triangle factors and $C_{3x}$-factors","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Adri\\'an Pastine, David Kamin, John Asplund, Melissa Keranen, Sibel \\\"Ozkan","submitted_at":"2015-10-15T16:20:42Z","abstract_excerpt":"The Hamilton-Waterloo Problem (HWP) in the case of $C_{m}$-factors and $C_{n}$-factors asks if $K_v$, where $v$ is odd (or $K_v-F$, where $F$ is a 1-factor and $v$ is even), can be decomposed into r copies of a 2-factor made either entirely of $m$-cycles and $s$ copies of a 2-factor made entirely of $n$-cycles. In this paper, we give some general constructions for such decompositions and apply them to the case where $m=3$ and $n=3x$. We settle the problem for odd $v$, except for a finite number of $x$ values. When $v$ is even, we make significant progress on the problem, although open cases ar"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1510.04607","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"}