{"paper":{"title":"Line k-Arboricity in Product Networks","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"He Li, Nan Jia, Yaping Mao, Zhiwei Guo","submitted_at":"2016-03-14T03:18:30Z","abstract_excerpt":"A \\emph{linear $k$-forest} is a forest whose components are paths of length at most $k$. The \\emph{linear $k$-arboricity} of a graph $G$, denoted by ${\\rm la}_k(G)$, is the least number of linear $k$-forests needed to decompose $G$. Recently, Zuo, He and Xue studied the exact values of the linear $(n-1)$-arboricity of Cartesian products of various combinations of complete graphs, cycles, complete multipartite graphs. In this paper, for general $k$ we show that $\\max\\{{\\rm la}_{k}(G),{\\rm la}_{\\ell}(H)\\}\\leq {\\rm la}_{\\max\\{k,\\ell\\}}(G\\Box H)\\leq {\\rm la}_{k}(G)+{\\rm la}_{\\ell}(H)$ for any two "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.04121","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"}