The spanning connectivity of line graphs

AbstractA k-container of G between u and v, C(u,v), is a set of k internally 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. Thus, every 1∗-connected graph is Hamiltonian connected. Moreover, every 2∗-connected graph is Hamiltonian. Zhan proved that G=L(M) is Hamiltonian connected if the edge-connectivity of M is at least 4. In this paper, we generalize this result by proving G=L(M) is k∗-connected if the edge-connectivity of M is at least max{2k,4}. We also generalize our result into spanning fan-connectivity