On extendability of co-edge-regular graphs

Let ℓ denote a non-negative integer. A connected graph Γ of even order at least 2ℓ+2 is ℓ-extendable if it contains a matching of size ℓ and if every such matching is contained in a perfect matching of Γ. A regular graph Γ is co-edge-regular if there exists a constant μ such that any pair of distinct nonadjacent vertices have μ common neighbors. In this paper we classify all 2-extendable and all 3-extendable co-edge-regular graphs of even order. Our results show that the only connected co-edge-regular graph of even order at least 8 and valence at least 7 which is not 3-extendable is the complete multipartite graph K4,4,4.