{"paper":{"title":"Cubic graphs with small independence ratio","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Alexandr Kostochka, J\\'ozsef Balogh, Xujun Liu","submitted_at":"2017-08-14T02:46:37Z","abstract_excerpt":"Let $i(r,g)$ denote the infimum of the ratio $\\frac{\\alpha(G)}{|V(G)|}$ over the $r$-regular graphs of girth at least $g$, where $\\alpha(G)$ is the independence number of $G$, and let $i(r,\\infty) := \\lim\\limits_{g \\to \\infty} i(r,g)$. Recently, several new lower bounds of $i(3,\\infty)$ were obtained. In particular, Hoppen and Wormald showed in 2015 that $i(3, \\infty) \\ge 0.4375$, and Cs\\'oka improved it to $i(3,\\infty) \\ge 0.44533$ in 2016. Bollob\\'as proved the upper bound $i(3,\\infty) < \\frac{6}{13}$ in 1981, and McKay improved it to $i(3,\\infty) < 0.45537$ in 1987. There were no improvemen"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.03996","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"}