New Results on EX Graphs

By the extremal number ex(n; t) = ex(n; {C ₃, C ₄, . . . , C _t }) we denote the maximum size (that is, number of edges) in a graph of order n > t and girth at least g ≥ t + 1. The set of all the graphs of order n, containing no cycles of length ≥ t, and of size ex(n; t), is denoted by EX(n; t) = EX(n; {C ₃, C ₄, . . . , C _t }), these graphs are called EX graphs. In 1975, Erdős proposed the problem of determining the extremal numbers ex(n; 4) of a graph of order n and girth at least 5. In this paper, we consider a generalized version of this problem, for t ≥ 5. In particular, we prove that ex(29; 6) = 45, also we improve some lower bounds and upper bounds of ex _u (n; t), for some particular values of n and t.