{"paper":{"title":"The $(k,\\ell)$-proper index of graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Colton Magnant, Hong Chang, Xueliang Li, Zhongmei Qin","submitted_at":"2016-06-13T09:30:48Z","abstract_excerpt":"A tree $T$ in an edge-colored graph is called a {\\it proper tree} if no two adjacent edges of $T$ receive the same color. Let $G$ be a connected graph of order $n$ and $k$ be an integer with $2\\leq k \\leq n$. For $S\\subseteq V(G)$ and $|S| \\ge 2$, an $S$-tree is a tree containing the vertices of $S$ in $G$. Suppose $\\{T_1,T_2,\\ldots,T_\\ell\\}$ is a set of $S$-trees, they are called \\emph{internally disjoint} if $E(T_i)\\cap E(T_j)=\\emptyset$ and $V(T_i)\\cap V(T_j)=S$ for $1\\leq i\\neq j\\leq \\ell$. For a set $S$ of $k$ vertices of $G$, the maximum number of internally disjoint $S$-trees in $G$ is "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.03872","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"}