# MathSciDoc: An Archive for Mathematician ∫

#### Combinatoricsmathscidoc:1612.06001

Journal of Combinatorial Mathematics and Combinatorial Computing, 91, 51-64, 2014.11
Let \$f(n)\$ be the maximum number of edges in a graph on \$n\$ vertices in which no two cycles have the same length. Erd\"{o}s raised the problem of determining \$f(n)\$. Erd\"{o}s conjectured that there exists a positive constant \$c\$ such that \$ex(n,C_{2k})\geq cn^{1+1/k}\$. Haj\'{o}s conjecture that every simple even graph on \$n\$ vertices can be decomposed into at most \$n/2\$ cycles. We present the problems, conjectures related to these problems and we summarize the know results. We do not think Haj\'{o}s conjecture is true.
Haj\'{o}s conjecture; even graph; Turan number; cycle; the maximum number of edges
```@inproceedings{lai2014some,
title={Some open problems on cycles},
author={Lai Chunhui, and Liu Mingjing},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161227083142622116692},
booktitle={Journal of Combinatorial Mathematics and Combinatorial Computing},
volume={91},
pages={51-64},
year={2014},
}
```
Lai Chunhui, and Liu Mingjing. Some open problems on cycles. 2014. Vol. 91. In Journal of Combinatorial Mathematics and Combinatorial Computing. pp.51-64. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20161227083142622116692.