A new bound for the connectivity of cages

An ( r , g ) -cage is an r -regular graph of girth g of minimum order. We prove that all ( r , g ) -cages are at least ⌈ r / 2 ⌉ -connected for every odd girth g ≥ 7 by means of a matrix technique which allows us to construct graphs without short cycles. This lower bound on the vertex connectivity of cages is a new advance in proving the conjecture of Fu, Huang and Rodger which states that all ( r , g ) -cages are r -connected.