On extremal bipartite graphs with high girth

Let us denote by E X ( m , n ; { C 4 , … , C 2 t } ) the family of bipartite graphs G with m and n vertices in its classes that contain no cycles of length less than or equal to 2t and have maximum size. In this paper the following question is proposed: does always such an extremal graph G contain a ( 2 t + 2 ) -cycle? The answer is shown to be affirmative for t = 2 , 3 or whenever m and n are large enough in comparison with t. The latter asymptotical result needs two preliminary theorems. First we state that the diameter of an extremal bipartite graph is at most 2t, and afterwards we show that its girth is equal to 2 t + 2 when the minimum degree is at least 2 and the maximum degree is at least t + 1 . We also give the exact value of the extremal function e x ( m , n ; { C 4 , … , C 2 t } ) for m = n = 2 t and m = n = 2 t + 1 and show that all the extremal bipartite graphs of E X ( m , n ; { C 4 , … , C 2 t } ) are maximally connected.