New families of graphs without short cycles and large size

We denote by ex(n; {C(3), C(4),..., C(s)}) or f(s) (n) the maximum number of edges in a graph of order n and girth at least s + 1. First we give a method to transform an n-vertex graph of girth g into a graph of girth at least g - 1 on fewer vertices. For an infinite sequence of values of n and s is an element of {4, 6, 10} the obtained graphs are denser than the known constructions of graphs of the same girth s + 1. We also give another different construction of dense graphs for an infinite sequence of values of n and s is an element of {7, 11}. These two methods improve the known lower bounds on f(s)(n) for s is an element of {4, 6, 7, 10, 11} which were obtained using different algorithms. Finally, to know how good are our results, we have proved that lim sup(n ->infinity) f(s)(n)/n(1+2/s-1) = 2(-1-2/s-1) for s is an element of {5, 7, 11}, and s(-1-2/s) <= lim sup(n ->infinity) f(s)(n)/n(1+2/s) <= 0.5 for s is an element of {6, 10(n)}. (C) 2010 Elsevier B.V. All rights reserved.