Maximum Degree of $3$-Independent Vertices and Group Connectivity

Abstract

Let $A$ be an abelian group with $|A|\ge 4$. Suppose that $G$ is a $3$-edge-connected simple graph on $n\ge 19$ vertices. We show in this paper that if $max\{d(x),d(y),d(z)\}\ge n/6$ 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., $30(2014),1055-1063).$