Some Open Problems on Cycles

Let \( f(n) \) be the maximum number of edges in a graph on \( n \) vertices in which no two cycles have the same length. Erdős raised the problem of determining \( f(n) \). Erdős conjectured that there exists a positive constant \( c \) such that \( ex(n, C_{2k}) \geq cn^{1+\frac{1}{k}} \). Hajós conjectured that every simple even graph on \( n \) vertices can be decomposed into at most \(\frac{n}{2}\) cycles. We present the problems, conjectures related to these problems, and we summarize the known results. We do not think Hajós’ conjecture is true.