Tree Connectivities of Cayley Graphs on Abelian Groups with Small Degrees

Abstract

The generalized k-connectivity κk(G) and the generalized k-edge-connectivity λk(G) of a graph G, also known as the tree connectivities, were introduced by Hager (J Comb Theory Ser B 38:179–189, 1985) and Li et al. (Discret Math Theor Comput Sci 14:43–54, 2012), respectively. In this paper, we study these invariants for Cayley graphs on Abelian groups with degree 3 or 4. When G is cubic, we prove κk(G) = λk(G) = 2 for 3 ≤ k≤ 6 and κk(G) = λk(G) = 1 for 7 ≤ k≤ n. When G has degree 4, we obtain κ3(G) = λ3(G) = 3 , λk(G) = 2 and κk(G) ≤ 2 for k≥ 8 , and κk(G) = 2 for k= n- 1 , n.