{"paper":{"title":"How many random edges make an almost-Dirac graph Hamiltonian?","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alberto Espuny D\\'iaz, Richarlotte Val\\'er\\`a Razafindravola","submitted_at":"2024-10-18T13:07:11Z","abstract_excerpt":"We study Hamiltonicity in the union of an $n$-vertex graph $H$ with high minimum degree and a binomial random graph on the same vertex set. In particular, we consider the case when $H$ has minimum degree close to $n/2$. We determine the perturbed threshold for Hamiltonicity in this setting.\n  To be precise, let $\\eta:= n/2-\\delta(H)$. For $\\eta=\\omega(1)$, we show that it suffices to add $\\Theta(\\eta)$ random edges to $H$ to a.a.s. obtain a Hamiltonian graph; for $\\eta=\\Theta(1)$, we show that $\\omega(1)$ edges suffice. In fact, when $\\eta=o(n)$ and $\\eta=\\omega(1)$, we show that $(8+o(1))\\eta"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2410.14447","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/2410.14447/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"}