On the girth of extremal graphs without shortest cycles

Let EX(@n;{C"3,...,C"n}) denote the set of graphs G of order @n that contain no cycles of length less than or equal to n which have maximum number of edges. In this paper we consider a problem posed by several authors: does G contain an n+1 cycle? We prove that the diameter of G is at most n-1, and present several results concerning the above question: the girth of G is g=n+1 if (i) @n>=n+5, diameter equal to n-1 and minimum degree at least 3; (ii) @n>=12, @n@?{15,80,170} and n=6. Moreover, if @n=15 we find an extremal graph of girth 8 obtained from a 3-regular complete bipartite graph subdividing its edges. (iii) We prove that if @n>=2n-3 and n>=7 the girth is at most 2n-5. We also show that the answer to the question is negative for @n@?n+1+@?(n-2)/2@?.