{"paper":{"title":"The Hamilton cycle space of random regular graphs and randomly perturbed graphs","license":"http://creativecommons.org/licenses/by-nc-nd/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Dan Hefetz, Michael Krivelevich","submitted_at":"2025-07-06T17:54:05Z","abstract_excerpt":"The cycle space of a graph $G$, denoted $C(G)$, is a vector space over ${\\mathbb F}_2$, spanned by all incidence vectors of edge-sets of cycles of $G$. If $G$ has $n$ vertices, then $C_n(G)$ is the subspace of $C(G)$, spanned by the incidence vectors of Hamilton cycles of $G$. We prove that asymptotically almost surely $C_n(G_{n,d}) = C(G_{n,d})$ holds whenever $n$ is odd and $d$ is a sufficiently large (even) integer. This extends (though with a weaker bound on $d$) the well-known result asserting that $G_{n,d}$ is asymptotically almost surely Hamiltonian for every $d \\geq 3$ (but not for $d "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2507.04488","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2507.04488/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"}