Maximum Generalized Local Connectivities of Cubic Cayley graphs on Abelian Groups

For a set \( S \) of \( k \) vertices of \( G \), let \( \kappa(S) \) denote the maximum number \( \ell \) of pairwise edge-disjoint trees \( T_1, T_2, \ldots, T_\ell \) in \( G \) such that \( V(T_i) \cap V(T_j) = S \) for \( 1 \leq i \neq j \leq \ell \) and \( \lambda(S) \) denote the maximum number \( \ell \) of pairwise edge-disjoint trees \( T_1, T_2, \ldots, T_\ell \) in \( G \) such that \( S \subseteq V(T_i) \) for \( 1 \leq i \leq \ell \). 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