{"paper":{"title":"Local resilience of an almost spanning $k$-cycle in random graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Angelika Steger, Milo\\v{s} Truji\\'c, Nemanja \\v{S}kori\\'c","submitted_at":"2017-09-12T15:12:22Z","abstract_excerpt":"The famous P\\'{o}sa-Seymour conjecture, confirmed in 1998 by Koml\\'{o}s, S\\'{a}rk\\\"{o}zy, and Szemer\\'{e}di, states that for any $k \\geq 2$, every graph on $n$ vertices with minimum degree $kn/(k + 1)$ contains the $k$-th power of a Hamilton cycle. We extend this result to a sparse random setting.\n  We show that for every $k \\geq 2$ there exists $C > 0$ such that if $p \\geq C(\\log n/n)^{1/k}$ then w.h.p. every subgraph of a random graph $G_{n, p}$ with minimum degree at least $(k/(k + 1) + o(1))np$, contains the $k$-th power of a cycle on at least $(1 - o(1))n$ vertices, improving upon the rec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1709.03901","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"}