{"paper":{"title":"The $k$-proper index of complete bipartite and complete multipartite graphs","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Jingshu Zhang, Wenjing Li, Xueliang Li","submitted_at":"2016-07-30T11:53:08Z","abstract_excerpt":"Let $G$ be a nontrivial connected graph of order $n$ with an edge-coloring $c:E(G)\\rightarrow\\{1,2,\\dots,t\\}$,$t\\in\\mathbb{N}$, where adjacent edges may be colored with the same color. A tree $T$ in $G$ is a \\emph{proper tree} if no two adjacent edges of it are assigned the same color. Let $k$ be a fixed integer with $2\\leq k\\leq n$. For a vertex subset $S\\subseteq V(G)$ with $|S|\\geq 2$, a tree is called an \\emph{$S$-tree} if it connects $S$ in $G$ . A \\emph{$k$-proper coloring} of $G$ is an edge-coloring of $G$ having the property that for every set $S$ of $k$ vertices of $G$, there exists a"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1608.00105","kind":"arxiv","version":3},"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"}