ABSTRACT
Assume that G is a k-connected graph. It follows from Menger’s Theorem that there are k internally vertex-disjoint (abbreviated as disjoint) paths joining any two distinct vertices u and v. A k-container C(u, v) of G is a set of k disjoint paths joining u to v. Here, we discuss another type of container, called spanning container. A spanning k-container, (abbreviated as k∗-container), C(u, v) is a k-container that contains all vertices of G. A graph G is k∗-connected if there exists a k∗-container between any two distinct vertices. In particular, a graph G is 1∗-connected if and only if it is Hamiltonian-connected, and a graphG is 2∗-connected if and only if it is Hamiltonian. All 1∗-connected graphs except that K1 and K2 are 2∗-connected. Thus, we defined the spanning connectivity of a graph G, κ∗(G), to be the largest integer k such that G is w∗-connected for all 1≤w≤ k if G is a 1∗-connected graph. Obviously, spanning connectivity is a hybrid concept of Hamiltonicity and connectivity. A graph G is super spanning connected if κ∗(G)= κ(G). Obviously, the complete graphKn is super spanning connected if n≥ 2.
