Graphs without repeated cycle lengths

Abstract

In 1975, P. Erd\"{o}s 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 $$f(n)\geq n+36t$$ for $t=1260r+169 (r\geq 1)$ and $n \geq 540t^{2}+{175811/2}t+{7989/2}$. Consequently, $\liminf\sb {n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + {2 \over 5}}.$ We make the following conjecture: \par \bigskip \noindent{\bf Conjecture.} $$\lim_{n \to \infty} {f(n)-n\over \sqrt n}=\sqrt {2.4}.$$