An old problem of Erd\H os: a graph without two cycles of the same length

Lai Chunhui Minnan Normal University

Combinatorics mathscidoc:2110.06001

arXiv:2110.04696, 2021.10
In 1975, P. Erd\H os proposed the problem of determining the maximum number $f(n)$ of edges in a graph on $n$ vertices in which any two cycles are of different lengths. Let $f^{\ast}(n)$ be the maximum number of edges in a simple graph on $n$ vertices in which any two cycles are of different lengths. Let $M_n$ be the set of simple graphs on $n$ vertices in which any two cycles are of different lengths and with the edges of $f^{\ast}(n)$. Let $mc(n)$ be the maximum cycle length for all $G \in M_n$. In this paper, it is proved that for $n$ sufficiently large, $mc(n)\leq \frac{15}{16}n.$ \par We make the following conjecture: \par \bigskip \noindent{\bf Conjecture.} $$\lim_{n \rightarrow \infty} {mc(n)\over n}= 0.$$
graph, cycle, number of edges
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@inproceedings{lai2021an,
  title={An old problem of Erd\H os: a graph without two cycles of the same length},
  author={Lai Chunhui},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20211013142626071808877},
  booktitle={arXiv:2110.04696},
  year={2021},
}
Lai Chunhui. An old problem of Erd\H os: a graph without two cycles of the same length. 2021. In arXiv:2110.04696. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20211013142626071808877.
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