Let $f(n)$ be the maximum number of edges in a graph on $n$ vertices
in which no two cycles have the same length. In 1975, Erdos
raised the problem of determining $f(n)$ (see J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (Macmillan, New York, 1976), p.247, Problem 11). Given a graph $H$, what is the maximum number of edges of a graph
with $n$ vertices not containing $H$ as a subgraph? This number is
denoted $ex(n,H)$, and is known as the Turan number. Erdos
conjectured that there exists a positive constant $c$ such that
$ex(n,C_{2k})\geq cn^{1+1/k}$(see P. Erdos, Some unsolved problems
in graph theory and combinatorial analysis, Combinatorial
Mathematics and its Applications (Proc. Conf., Oxford, 1969) , pp.
97--109, Academic Press, London, 1971). Haj\'{o}s conjectured that
every simple even graph on $n$ vertices can
be decomposed into at most $n/2$ cycles (see L. Lovasz, On
covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of
Graphs, Academic Press, New York, 1968, pp. 231 - 236 ). We present the problems,
conjectures related to these problems and we summarize the know results.