On the spanning connectivity of graphs

A k-container  C(u,v)C(u,v) of G   between uu and vv is a set of k   internally disjoint paths between uu and vv. A k  -container C(u,v)C(u,v) of G   is a k*k*-container if it contains all vertices of G. A graph G   is k*k*-connected   if there exists a k*k*-container between any two distinct vertices. The spanning connectivity of G  , κ*(G)κ*(G), is defined to be the largest integer k such that G   is w*w*-connected for all 1⩽w⩽k1⩽w⩽k if G   is a 1*1*-connected graph. In this paper, we prove that κ*(G)⩾2δ(G)-n(G)+2κ*(G)⩾2δ(G)-n(G)+2 if (n(G)/2)+1⩽δ(G)⩽n(G)-2(n(G)/2)+1⩽δ(G)⩽n(G)-2. Furthermore, we prove that κ*(G-T)⩾2δ(G)-n(G)+2-|T|κ*(G-T)⩾2δ(G)-n(G)+2-|T| if T   is a vertex subset with |T|⩽2δ(G)-n(G)-1|T|⩽2δ(G)-n(G)-1.