{"paper":{"title":"Solutions to conjectures on the $(k,\\ell)$-rainbow index of complete graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jiangli Song, Qingqiong Cai, Xueliang Li","submitted_at":"2012-12-31T09:24:43Z","abstract_excerpt":"The $(k,\\ell)$-rainbow index $rx_{k, \\ell}(G)$ of a graph $G$ was introduced by Chartrand et. al. For the complete graph $K_n$ of order $n\\geq 6$, they showed that $rx_{3,\\ell}(K_n)=3$ for $\\ell=1,2$. Furthermore, they conjectured that for every positive integer $\\ell$, there exists a positive integer $N$ such that $rx_{3,\\ell}(K_{n})=3$ for every integer $n \\geq N$. More generally, they conjectured that for every pair of positive integers $k$ and $\\ell$ with $k\\geq 3$, there exists a positive integer $N$ such that $rx_{k,\\ell}(K_{n})=k$ for every integer $n \\geq N$. This paper is to give solu"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.6845","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"}