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

Abstract

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 1a parts per thousand currency signka parts per thousand currency sign kappa(G) where kappa(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 1a parts per thousand currency signka parts per thousand currency sign kappa(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.