{"paper":{"title":"Proof of the $1$-factorization and Hamilton Decomposition Conjectures","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Allan Lo, Andrew Treglown, B\\'ela Csaba, Daniela K\\\"uhn, Deryk Osthus","submitted_at":"2014-01-16T20:44:15Z","abstract_excerpt":"In this paper we prove the following results (via a unified approach) for all sufficiently large $n$:\n  (i) [$1$-factorization conjecture] Suppose that $n$ is even and $D\\geq 2\\lceil n/4\\rceil -1$. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into perfect matchings. Equivalently, $\\chi'(G)=D$.\n  (ii) [Hamilton decomposition conjecture] Suppose that $D \\ge \\lfloor n/2 \\rfloor $. Then every $D$-regular graph $G$ on $n$ vertices has a decomposition into Hamilton cycles and at most one perfect matching.\n  (iii) [Optimal packings of Hamilton cycles] Suppose that $G$ is a gra"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1401.4159","kind":"arxiv","version":2},"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"}