{"paper":{"title":"On a Conjecture of Erd\\H{o}s, Gallai, and Tuza","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Gregory J. Puleo","submitted_at":"2013-11-21T08:36:27Z","abstract_excerpt":"Erd\\H{o}s, Gallai, and Tuza posed the following problem: given an $n$-vertex graph $G$, let $\\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\\alpha_1(G)$ denote the largest size of a set of edges containing at most one edge from each triangle of $G$. Is it always the case that $\\alpha_1(G) + \\tau_1(G) \\leq n^2/4$? We have two main results. We first obtain the upper bound $\\alpha_1(G) + \\tau_1(G) \\leq 5n^2/16$, as a partial result towards the Erd\\H{o}s--Gallai--Tuza conjecture. We also show that always $\\alpha_1(G) \\leq n^2/2 - m$, where $m"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1311.5332","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"}