{"paper":{"title":"Proper connection number and graph products","license":"http://creativecommons.org/licenses/by/3.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Chengfu Ye, Fengnan Yanling, Yaping Mao, Zhao Wang","submitted_at":"2015-05-09T08:56:39Z","abstract_excerpt":"A path $P$ in an edge-colored graph $G$ is called \\emph{a proper path} if no two adjacent edges of $P$ are colored the same, and $G$ is \\emph{proper connected} if every two vertices of $G$ are connected by a proper path in $G$. The \\emph{proper connection number} of a connected graph $G$, denoted by $pc(G)$, is the minimum number of colors that are needed to make $G$ proper connected. In this paper, we study the proper connection number on the lexicographical, strong, Cartesian, and direct product and present several upper bounds for these products of graphs."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1505.02246","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"}