image-cages with image are 4-connected Original Research Article

A regular graph G of degree δδ and girth g is said to be a (δ,g)(δ,g)-cage if it has the least number of vertices among all δδ-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)(δ,g)-cage is 4-connected provided that either δ=4δ=4, or δ⩾5δ⩾5 and g⩾10g⩾10. These results support the conjecture of Fu, Huang and Rodger that all (δ,g)(δ,g)-cages are δδ-connected.