The spanning connectivity of line graphs

A kk-container   of GG between uu and vv, C(u,v)C(u,v), is a set of kk internally disjoint paths between uu and vv. A k∗k∗-container  C(u,v)C(u,v) of GG is a kk-container if it contains all vertices of GG. A graph GG is k∗k∗-connected   if there exists a k∗k∗-container between any two distinct vertices. Thus, every 1∗1∗-connected graph is Hamiltonian connected. Moreover, every 2∗2∗-connected graph is Hamiltonian. Zhan proved that G=L(M)G=L(M) is Hamiltonian connected if the edge-connectivity of MM is at least 4. In this paper, we generalize this result by proving G=L(M)G=L(M) is k∗k∗-connected if the edge-connectivity of MM is at least max{2k,4}max{2k,4}. We also generalize our result into spanning fan-connectivity.