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

Abstract

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 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 P n is super connected if and only if n ≠ 3 and the n-dimensional star graph S n is super laceable if and only if n ≠ 3 .