{"paper":{"title":"On the number of $K_4$-saturating edges","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Liu, J\\'ozsef Balogh","submitted_at":"2013-12-18T18:00:48Z","abstract_excerpt":"Let $G$ be a $K_4$-free graph, an edge in its complement is a $K_4$-\\emph{saturating} edge if the addition of this edge to $G$ creates a copy of $K_4$. Erd\\H{o}s and Tuza conjectured that for any $n$-vertex $K_4$-free graph $G$ with $\\lfloor n^2/4\\rfloor+1$ edges, one can find at least $(1+o(1))\\frac{n^2}{16}$ $K_4$-saturating edges. We construct a graph with only $\\frac{2n^2}{33}$ $K_4$-saturating edges. Furthermore, we prove that it is best possible, i.e., one can always find at least $(1+o(1))\\frac{2n^2}{33}$ $K_4$-saturating edges in an $n$-vertex $K_4$-free graph with $\\lfloor n^2/4\\rfloo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1312.5248","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"}