Connectivity of cages

A (k; g)-graph is a k-regular graph with girth g. Let f(k; g) be the smallest integer ν such there exists a (k; g)-graph with ν vertices. A (k; g)-cage is a (k; g)-graph with f(k; g) vertices. In this paper we prove that the cages are monotonic in that f(k; g1) < f(k; g2) for all k ≥ 3 and 3 ≤ g1 < g2. We use this to prove that (k; g)-cages are 2-connected, and if k = 3 then their connectivity is k. c © 1997 John Wiley & Sons, Inc.