The super spanning connectivity and super spanning laceability of the enhanced hypercubes

A k -container C ( u , v ) of a graph G is a set of k disjoint paths between u and v . A k -container C ( u , v ) of G is a k * -container if it contains all vertices of G . A graph G is k * -connected if there exists a k * -container between any two distinct vertices of G . Therefore, a graph is 1 * -connected (respectively, 2 * -connected) if and only if it is Hamiltonian connected (respectively, Hamiltonian). A graph G is super spanning connected if there exists a k * -container between any two distinct vertices of G for every k with 1≤ k ≤ κ ( G ) where κ ( G ) is the connectivity of G . A bipartite graph G is k * -laceable if there exists a k * -container between any two vertices from different partite set of G . A bipartite graph G is super spanning laceable if there exists a k * -container between any two vertices from different partite set of G for every k with 1≤ k ≤ κ ( G ). In this paper, we prove that the enhanced hypercube Q n , m is super spanning laceable if m is an odd integer and super spanning connected if otherwise.