Every cubic cage is quasi 4-connected

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 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.