, Volume 29, Issue 1, pp 105-119

On Superconnectivity of (4, g)-Cages

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Abstract

A (k, g)-cage is a graph that has the least number of vertices among all k-regular graphs with girth g. It has been conjectured (Fu et al. in J. Graph Theory, 24:187–191, 1997) that all (k, g)-cages are k-connected for every k ≥ 3. A k-connected graph G is called superconnected if every k-cutset S is the neighborhood of some vertex. Moreover, if GS has precisely two components, then G is called tightly superconnected. In this paper, we prove that every (4, g)-cage is tightly superconnected when g ≥ 11 is odd.

This work was supported by the National Natural Science Foundation of China (No. 11101329).