Maximum generalized local connectivities of cubic Cayley graphs on Abelian groups

For a set S of k vertices of G1 let κ(S) denote the maximum number of pairwise edge-disjoint trees T1, T2,, Tl in G such that V(Ti) ∩ V(Tj) = S for 1 ≤ i ≈ j ≤ and λ(S) denote the maximum number of pairwise edge-disjoint trees T1, T2,, Tl in G such that S ⊆ V(Ti) for 1 ≤ i ≤ . Similar to the classical maximum local connectivity, H. Li et al. introduced the parameter κk(G) = max(κ(S)|S ⊆ 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 λk(G) = max(λ(S)|S ⊆ V(G), |S| = k). In this paper, we investigate the maximum generalized local connectivity and edgeconnectivity of a cubic connected Cayley graph G on an Abelian group. We determine the precise values for κ3(G) and λ3(G) and also prove some results of κk(G) and λk(G) for general k.