Decreasing the diameter of cycles

Alon, Gyárfás and Ruszinkó [1] established a bound for the minimum number of edges that have to be added to a graph of n ≥ n0 vertices in order to obtain a graph of diameter 2. Alon et al. approximated n0 by a polynomial function of the maximum degree of the initial graph. They conjectured that the minimum value for n0 is 12 for the case of Cn, as opposed to n0 = 274 obtained by calculations for the general case. In this paper we prove their conjecture. For the reduction to diameter 3 of a cycle we also improve the bounds from [1] showing that the minimum number of edges that need to be added to the cycle Cn is between n− 59 and n− 8.