{"paper":{"title":"Triangle-independent sets vs. cuts","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Sergey Norin, Yue Ru Sun","submitted_at":"2016-02-13T19:54:20Z","abstract_excerpt":"A set of edges $T$ in a graph $G$ is triangle-independent if $T$ contains at most one edge from each triangle in $G$. Let $\\alpha_1(G)$ denote the maximum size of the triangle-independent set in $G$, and let $\\tau_B(G)$ denote minimum size of a set $F \\subseteq E(G)$ such that $G \\setminus F$ is bipartite. We prove that $$\\alpha_1(G) + \\tau_B(G) \\leq \\frac{|V(G)|^2}{4},$$ verifying a conjecture due to Lehel, and independently Puleo, and a slightly weaker conjecture of Erd\\H{o}s, Gallai and Tuza. Further, we characterize the graphs which attain the equality."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1602.04370","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":""},"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"}