Lower bound of cyclic edge connectivity for n-extendability of regular graphs

Abstract A cyclically m-edge-connected n-connected k-regular graph is called an (m.n.k) graph. It is proved that for any m > 0 and k ⩾3, there is an (m, k, k) bipartite graph. A graph G is n-extendable if every matching of size n in G lies in a perfect matching of G. We prove the existence of a (k2 −1, k + 1, k + 1) bipartite graph which is not k-extendable and the existence of an (m, k + 1, k + 1) graph which is not n-extendable, where n ⩾ 2, k ⩾ 2 and m is any positive integer. The existence of the former graphs shows that a result of Holton and Plummer is sharp.