{"paper":{"title":"Proper vertex connection and graph operations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Xiaoyu Zhu, Yingying Zhang","submitted_at":"2017-05-06T14:12:11Z","abstract_excerpt":"A path in a vertex-colored graph is a {\\it vertex-proper path} if any two internal adjacent vertices differ in color. A vertex-colored graph is {\\it proper vertex $k$-connected} if any two vertices of the graph are connected by $k$ disjoint vertex-proper paths of the graph. For a $k$-connected graph $G$, the {\\it proper vertex $k$-connection number} of $G$, denoted by $pvc_{k}(G)$, is defined as the smallest number of colors required to make $G$ proper vertex $k$-connected. A vertex-colored graph is {\\it strong proper vertex-connected}, if for any two vertices $u,v$ of the graph, there exists "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1705.02486","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"}