The generalized 4-connectivity of a family of regular graphs

Let G be a connected graph and S ⊆ V ( G ) with | S | ≥ 2 . A tree T in G is called an S -tree if S ⊆ V ( T ) . Two S -trees T 1 and T 2 are called internally disjoint if E ( T 1 ) ∩ E ( T 2 ) = 0̸ and V ( T 1 ) ∩ V ( T 2 ) = S . For an integer k with 2 ≤ k ≤ n , the generalized k -connectivity of a graph G is defined as κ k ( G ) = min { κ G ( S ) | S ⊆ V ( G ) and | S | = k } , where κ G ( S ) denotes the maximum number of internally disjoint S -trees in G . The generalized k -connectivity extends traditional connectivity and serves as a crucial measure for evaluating the reliability and fault tolerance of connecting any k vertices in G . In this paper, we mainly investigate the generalized 4-connectivity of a family of regular graph G n , which improves the known results about generalized 3-connectivity of G n in Zhao et al., (2021). For a reason that the alternating group network A N n , the star graph S n and pancake graphs P n are special cases of the regular graph G n , as applications of the main result, we obtain that κ 4 ( A N n ) = n − 2 for n ≥ 4 , κ 4 ( S n ) = n − 2 for n ≥ 3 and κ 4 ( P n ) = n − 2 for n ≥ 3 .