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

Abstract

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 kappa(G) be the connectivity of G. A graph G is super connected if G is i*-connected for all 1 <= i <= kappa(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 <= kappa(G). In this paper, we prove that the n-dimensional pancake graph P(n) is super connected if and only if n not equal 3 and the n-dimensional star graph S(n) is super laceable if and only if n not equal 3. (c) 2005 Elsevier B.V. All rights reserved.