Fault-tolerant strong Menger (edge) connectivity of arrangement graph

Abstract If any two different vertices u , v in a connected graph G are connected by min { deg G ( u ) , deg G ( v ) } vertex (edge)-disjoint paths, then G is called strongly Menger (edge) connected. In this paper, we focus on discussing the strong Menger connectivity of the arrangement graph with faults. Specifically, we show that A n , k − S is strongly Menger connected for | S | ≤ ( k − 1 ) ( n − k ) − 1 where S ⊆ V ( G ) . Moreover, we show that A n , k − S is strongly Menger edge connected for | S | ≤ k ( n − k ) − 2 where S ⊆ E ( G ) . Further more, we show that, under the restricted condition δ ( A n , k − S ) ≥ 2 , A n , k − S is strongly Menger edge connected for | S | ≤ 2 k ( n − k ) − 6 where S ⊆ E ( G ) . The bounds above are all sharp.