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

A (k, g)-cage is a k-regular graph with girth g that has the fewest number of vertices. It has been conjectured [Fu, Huang, and Rodger, Connectivity of cages, 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, Wang, and Wang, On the connectivity of (4, g)-cages, Ars Combin 64 (2002), 181-192] that every (4, g)-cage is 4-connected. In this paper, we prove that every (4, g)-cage is tightly superconnected when g is even and g ≥ 12.