Connectivity of cages

Abstract

A (k; g)-graph is a k-regular graph with girth g. Let f(k; g) be the smallest integer nu such there exists a (k; g)-graph with nu 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; g(1)) < f(k; g(2)) for all k greater than or equal to 3 and 3 less than or equal to g(1) < g(2). 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.