The spanning connectivity of line graphs

A 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 { 2 k , 4 } . We also generalize our result into spanning fan-connectivity.