On superconnectivity of (4,g)-cages with even girth

Abstract

A (k, g)-cage is a k-regular graph with girth g that has the fewest number of vertices. It has been conjectured (Fu et al., J Graph Theory 24 (1997), 187–191) that all (k, g)-cages are k-connected for k ≥ 3. A connected graph G is said to be superconnected if every minimum cut-set S is the neighborhood of a vertex of minimum degree. Moreover, if G-S has precisely two components, then G is called tightly superconnected. It was shown (Xu et al., Ars Combin 64 (2002), 181–192) that every (4, g)-cage is 4-connected. In this article, we prove that every (4, g)-cage is tightly superconnected when g is even and g ≥ 12. © 2009 Wiley Periodicals, Inc. NETWORKS, 2010