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 λ > 1 are bk/3c-extendable (when μ 6 k/2) and d 4 e-extendable (when μ > k/2), where λ is the number of common neighbors of any two adjacent vertices and μ 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 dk/2e-extendable. We show that the extendability of many strongly regular graphs of valency k is at least dk/2e − 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.