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

A k-container C(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 (G) be the connectivity of G. A graph G is super connected if G is i∗-connected for all 1 i (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 (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. © 2005 Elsevier B.V. All rights reserved.