Maximum Degree of \(3\)-Independent Vertices and Group Connectivity

Abstract

Let \(A\) be an abelian group with \(|A| \geq 4\). Suppose that \(G\) is a \(3\)-edge-connected simple graph on \(n \geq 19\) vertices. We show in this paper that if \(\max\{d(x), d(y), d(z)\} \geq n/6\) for every \(3\)-independent vertices \(\{x, y, z\}\) of \(G\), then either \(G\) is \(A\)-connected or \(G\) can be \(T\)-reduced to the Petersen graph, which generalizes the result of Zhang and Li (Graphs and Combin., \(30 (2014), 1055-1063).\)