{"paper":{"title":"Degree sum conditions for graphs to have proper connection number 2","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.CO","authors_text":"Hong Chang, Xueliang Li, Zhong Huang","submitted_at":"2016-11-29T06:02:28Z","abstract_excerpt":"A path $P$ in an edge-colored graph $G$ is a \\emph{proper path} if no two adjacent edges of $P$ are colored with the same color. The graph $G$ is \\emph{proper connected} if, between every pair of vertices, there exists a proper path in $G$. The \\emph{proper connection number} $pc(G)$ of a connected graph $G$ is defined as the minimum number of colors to make $G$ proper connected. In this paper, we study the degree sum condition for a general graph or a bipartite graph to have proper connection number 2. First, we show that if $G$ is a connected noncomplete graph of order $n\\geq 5$ such that $d"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1611.09500","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"}