An old problem of Erd\H{o}s: a graph without two cycles of the same length

TL;DR

This paper investigates the maximum number of edges in graphs with all cycles of distinct lengths, proving an upper bound on the maximum cycle length relative to the number of vertices, and conjecturing this ratio tends to zero.

Contribution

It establishes an upper bound on the maximum cycle length in graphs with all cycles of different lengths and proposes a conjecture about the asymptotic behavior of this ratio.

Findings

Proved that for large n, the maximum cycle length mc(n) is at most (15/16)n.

Formulated a conjecture that the ratio mc(n)/n tends to zero as n grows.

Provides bounds and insights into Erdős's problem on cycle lengths in graphs.

Abstract

In 1975, P. Erd\H{o}s 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, $mc(n)\leq \frac{15}{16}n$. We make the following conjecture: $$\lim_{n \rightarrow \infty} {mc(n)\over n}= 0.$$