The Diameter of a Cycle Plus a Random Matching

How small can the diameter be made by adding a matching to an n-cycle? In this paper this question is answered by showing that the graph consisting of an n-cycle and a random matching has diameter about $\log _2 n$, which is very close to the best possible value. It is also shown that by adding a random matching to graphs with certain expanding properties such as expanders or Ramanujan graphs, the resulting graphs have near optimum diameters.