The super-connected property of recursive circulant graphs

In a graph G, a k-container Ck(u, v) is a set of k disjoint paths joining u and v. A k-container Ck(u, v) is k ∗-container if every vertex of G is passed by some path in Ck(u, v). A graph G is k ∗-connected if there exists a k∗-container between any two vertices. An m-regular graph G is super-connected if G is k∗-connected for any k with 1 k m. In this paper, we prove that the recursive circulant graphs G(2m,4), proposed by Park and Chwa [Theoret. Comput. Sci. 244 (2000) 35–62], are super-connected if and only if m = 2.  2004 Elsevier B.V. All rights reserved.