The extendability of matchings in strongly regular graphs

A graph G of even order v is called t-extendable if it contains a perfect matching, t < v/2 and any matching of t edges is contained in some perfect matching. The extendability of G is the maximum t such that G is t-extendable. In this paper, we study the extendability properties of strongly regular graphs. We improve previous results and classify all strongly regular graphs that are not 3-extendable. We also show that strongly regular graphs of valency k,- 3 with A,- 1 are [k/3] -extendable (when it,, k/2) and [kV-extendable (when it > k/2), where A is the number of common neighbors of any two adjacent vertices and it is the number of common neighbors of any two non-adjacent vertices. Our results are close to being best possible as there are strongly regular graphs of valency k that are not [k/21-extendable. We show that the extendability of many strongly regular graphs of valency k is at least [k/21 1 and we conjecture that this is true for all primitive strongly regular graphs. We obtain similar results for strongly regular graphs of odd order.