Formula omitted]-cages with [formula omitted] are 4-connected

Abstract

A regular graph G of degree δ and girth g is said to be a ( δ , g ) - cage if it has the least number of vertices among all δ -regular graphs with girth g. A graph is called k- connected if the order of every cutset is at least k. In this work, we prove that every ( δ , g ) -cage is 4-connected provided that either δ = 4 , or δ ⩾ 5 and g ⩾ 10 . These results support the conjecture of Fu, Huang and Rodger that all ( δ , g ) -cages are δ -connected.