New families of graphs without short cycles and large size

We denote by ex(n;{C3,C4,…,Cs})ex(n;{C3,C4,…,Cs}) or fs(n)fs(n) the maximum number of edges in a graph of order nn and girth at least s+1s+1. First we give a method to transform an nn-vertex graph of girth gg into a graph of girth at least g−1g−1 on fewer vertices. For an infinite sequence of values of nn and s∈{4,6,10}s∈{4,6,10} the obtained graphs are denser than the known constructions of graphs of the same girth s+1s+1. We also give another different construction of dense graphs for an infinite sequence of values of nn and s∈{7,11}s∈{7,11}. These two methods improve the known lower bounds on fs(n)fs(n) for s∈{4,6,7,10,11}s∈{4,6,7,10,11} which were obtained using different algorithms. Finally, to know how good are our results, we have proved that lim supn→∞fs(n)n1+2s−1=2−1−2s−1 for s∈{5,7,11}s∈{5,7,11}, and s−1−2s≤lim supn→∞fs(n)n1+2s≤0.5 for s∈{6,10}s∈{6,10}.