Cycles in graphs of fixed girth with large size

Consider a family of graphs having a fixed girth and a large size. We give an optimal lower asymptotic bound on the number of even cycles of any constant length, as the order of the graphs tends to infinity.

[1]  Benny Sudakov,et al.  Cycle lengths in sparse graphs , 2007, Comb..

[2]  W. G. Brown.  On Graphs that do not Contain a Thomsen Graph , 1966, Canadian Mathematical Bulletin.

[3]  M. Simonovits,et al.  Cycles of even length in graphs , 1974 .

[4]  Béla Bollobás,et al.  Extremal problems in graph theory , 1977, J. Graph Theory.

[5]  Christopher M. Hartman.  Extremal problems in graph theory , 1997 .

[6]  Boris Bukh,et al.  A Bound on the Number of Edges in Graphs Without an Even Cycle , 2014, Combinatorics, Probability and Computing.

[7]  C. T. Benson.  Minimal Regular Graphs of Girths Eight and Twelve , 1966, Canadian Journal of Mathematics.

[8]  I. Reiman.  Über ein Problem von K. Zarankiewicz , 1958 .

[9]  Peter Volkmann,et al.  Über ein Problem von Fenyő , 1984 .

[10]  David Saxton,et al.  The number of $C_{2l}$-free graphs , 2013, 1309.2927.

[11]  Rephael Wenger,et al.  Extremal graphs with no C4's, C6's, or C10's , 1991, J. Comb. Theory, Ser. B.