Some open problems on cycles

Lai Chunhui Minnan Normal University Liu Mingjing Minnan Normal University

Combinatorics mathscidoc: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
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@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.
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