The super connectivity of the pancake graphs and the super laceability of the star graphs

A k-containerC(u,v) of a graph G is a set of k-disjoint paths joining u to v. A k-container C(u,v) of G is a k*-container if it contains all the vertices of G. A graph G is k*-connected if there exists a k*-container between any two distinct vertices. Let κ(G) be the connectivity of G. A graph G is super connected if G is i*-connected for all 1⩽i⩽κ(G). A bipartite graph G is k*-laceable if there exists a k*-container between any two vertices from different parts of G. A bipartite graph G is super laceable if G is i*-laceable for all 1⩽i⩽κ(G). In this paper, we prove that the n-dimensional pancake graph Pn is super connected if and only if n≠3 and the n-dimensional star graph Sn is super laceable if and only if n≠3.