The codiameter of a 2-connected graph

The codiameter of a graph is defined as the minimum, taken over all pairs of vertices u   and vv in the graph, of the maximum length of a (u,v)(u,v)-path. A result of Fan [Long cycles and the codiameter of a graph, I, J. Combin. Theory Ser. B 49 (1990) 151–180.] is that, for an integer c⩾3c⩾3, if G is a 2-connected graph on n   vertices with more than ((c+1)/2)(n-2)+1((c+1)/2)(n-2)+1 edges, the codiameter of G is at least c  . The result is best possible when n-2n-2 is divisible by c-2c-2. In this paper, we shall show that the bound ((c+1)/2)(n-2)+1((c+1)/2)(n-2)+1 can be decreased when n-2n-2 is not divisible by c-2c-2.