{"paper":{"title":"A randomized construction of high girth regular graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Michael Simkin, Nati Linial","submitted_at":"2019-11-21T17:52:58Z","abstract_excerpt":"We describe a new random greedy algorithm for generating regular graphs of high girth: Let $k\\geq 3$ and $c \\in (0,1)$ be fixed. Let $n \\in \\mathbb{N}$ be even and set $g = c \\log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$ vertices. As long as the smallest degree $\\delta (G)<k$, choose, uniformly at random, two vertices $u,v \\in V(G)$ of degree $\\delta(G)$ whose distance is at least $g-1$. If there are no such vertex pairs, abort. Otherwise, add the edge $uv$ to $E(G)$.\n  We show that with high probability this algorithm yields a $k$-regular graph with girth at least $g$. Our analysis "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1911.09640","kind":"arxiv","version":3},"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/1911.09640/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"}