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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 ofan n-cycle and a random matching has diameter about log2n, 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.
Bollobás et al. (Mon,) studied this question.