On the size of graphs without repeated cycle lengths

TL;DR

This paper improves lower bounds on the maximum number of edges in graphs with all cycles of different lengths, providing new asymptotic results that surpass previous bounds and disprove a conjecture about their limit behavior.

Contribution

It establishes new lower bounds for the maximum edges in such graphs, refining the asymptotic understanding of Erdős's cycle length problem.

Findings

New lower bounds for f(n) in graphs with distinct cycle lengths.

Asymptotic limit of (f(n)-n)/√n exceeds previous bounds.

Disproof of the conjecture that the limit equals √2.4.

Abstract

In 1975, P. Erd\"os proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+\frac{107}{3}t+\frac{7}{3}$$ for $t=1260r+169 \,\ (r\geq 1)$ and $n \geq \frac{2119}{4}t^{2}+87978t+\frac{15957}{4}$. Consequently, $\lim \inf_{n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{7654}{19071}},$ which is better than the previous bounds $\sqrt 2$ [Y. Shi, Discrete Math. 71(1988), 57-71], $\sqrt {2.4}$ [C. Lai, Australas. J. Combin. 27(2003), 101-105]. The conjecture $\lim_{n \rightarrow \infty} {f(n)-n\over \sqrt n}=\sqrt {2.4}$ is not true.