Every cubic cage is quasi 4-connected Original Research Article

A (δ,g)-cage is a regular graph of degree δ and girth g with the least possible number of vertices. It was proved by Fu, Huang and Rodger that every (3,g)-cage is 3-connected. Moreover, the same authors conjectured that all (δ,g)-cages are δ-connected for every δ⩾3. As a first step towards the proof of this conjecture, Jiang and Mubayi, and independently Daven and Rodger, showed that every (δ,g)-cage with δ⩾3 is 3-connected. A 3-connected graph G is called quasi 4-connected if for each cutset S⊂V(G) with |S|=3, S is the neighbourhood of a vertex of degree 3 and G−S has precisely two components. In this paper, we prove that every (3,g)-cage with g⩾5 is quasi 4-connected, which can be seen as a further step towards the proof of the aforementioned conjecture.