{"paper":{"title":"Cycle-factors of regular graphs via entropy","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.DM","cs.DS","math.PR"],"primary_cat":"math.CO","authors_text":"Alp M\\\"uyesser, Ant\\'onio Gir\\~ao, Eoin Hurley, Lukas Michel, Micha Christoph, Nemanja Dragani\\'c","submitted_at":"2025-07-25T16:36:08Z","abstract_excerpt":"It is a classical result that a random permutation of $n$ elements has, on average, about $\\log n$ cycles. We generalise this fact to all directed $d$-regular graphs on $n$ vertices by showing that, on average, a random cycle-factor of such a graph has $\\mathcal{O}((n\\log d)/d)$ cycles. This is tight up to the constant factor and improves the best previous bound of the form $\\mathcal{O}(n/\\sqrt{\\log d})$ due to Vishnoi. Our results also yield randomised polynomial-time algorithms for finding such a cycle-factor and for finding a tour of length $(1+\\mathcal{O}((\\log d)/d)) \\cdot n$ if the graph"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.19417","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.19417/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}