  Hajos' conjecture that every simple even graph on $n$ vertices can be
decomposed into at most $(n-1)/2$ 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). Let $f(n)$ be the maximum number of edges in a graph on
$n$ vertices in which no two cycles have the same length. P. 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. P. 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). This paper summarizes some results on
these problems and the conjectures that relate to these. We do not think
Haj\'{o}s conjecture is true.
