Even Cycles in Graphs with Many Odd Cycles

Abstract.

It will be shown that if G is a graph of order n which contains a triangle, a cycle of length n or n−1 and at least cn odd cycles of different lengths for some positive constant c, then there exists some positive constant k=k(c) such that G contains at least kn ^1/6 even cycles of different lengths. Other results on the number of even cycle lengths which appear in graphs with many different odd length cycles will be given.