Maximum Generalized Local Connectivities of Cubic Cayley Graphs on Abelian Groups

Abstract

For a set $S$ of $k$ vertices of $G$, let $\kappa (S)$ denote the maximum number $\ell$ of pairwise edge-disjoint trees $T_1,T_2,\dots ,T_{\ell }$ in $G$ such that $V(T_i)\cap V(T_j)=S$ for $1\le i\ne j\le \ell$ and $\lambda (S)$ denote the maximum number $\ell$ of pairwise edge-disjoint trees $T_1,T_2,\dots ,T_{\ell }$ in $G$ such that $S\subseteq V(T_i)$ for $1\le i\le \ell$. Similar to the classical maximum local connectivity, H. Li et al. introduced the parameter ${\overline{\kappa }}_k(G)=max\{\kappa (S)\mid S\subseteq V(G),|S|=k\}$, which is called the maximum generalized local connectivity of $G$. The maximum generalized local edge-connectivity of $G$ which was introduced by X. Li et al. is defined as ${\overline{\lambda }}_k(G)=max\{\lambda (S)\mid S\subseteq V(G),|S|=k\}$. In this paper, we investigate the maximum generalized local connectivity and edge-connectivity of a cubic connected Cayley graph $G$ on an Abelian group. We determine the precise values for ${\overline{\kappa }}_3(G)$ and ${\overline{\lambda }}_3(G)$ and also prove some results of ${\overline{\kappa }}_k(G)$ and ${\overline{\lambda }}_k(G)$ for general $k$.