On the number of edges in some graphs

TL;DR

This paper investigates the maximum number of edges in graphs with constraints on cycle lengths, improving bounds on the difference between edges and vertices as the graph size grows.

Contribution

It establishes new lower bounds for the asymptotic difference between edges and vertices in graphs with cycle length restrictions, refining previous results.

Findings

Proves that lim inf of (f(n)-n)/sqrt(n) exceeds sqrt(2 + 40/99)

Shows that lim inf of (g(n,m)-n)/sqrt(n/m) exceeds sqrt(2.444) for even m

Conjectures a similar bound for the general function f(n) as n approaches infinity

Abstract

In 1975, P. Erd\H{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph with $n$ vertices in which any two cycles are of different lengths. The sequence $(c_1,c_2,\cdots,c_n)$ is the cycle length distribution of a graph $G$ of order $n$ where $c_i$ is the number of cycles of length $i$ in $G$. Let $f(a_1,a_2,\cdots, a_n)$ denote the maximum possible number of edges in a graph which satisfies $c_i\leq a_i$ where $a_i$ is a nonnegative integer. In 1991, Shi posed the problem of determining $f(a_1,a_2,\cdots,a_n)$ which extended the problem due to Erd\H{o}s, it is clear that $f(n)=f(1,1,\cdots,1)$. Let $g(n,m)=f(a_1,a_2,\cdots,a_n),$ $a_i=1$ for all $i/m$ be integer, $a_i=0$ for all $i/m$ be not integer. It is clear that $f(n)=g(n,1)$. We prove that $\liminf_{n \to \infty} {f(n)-n \over \sqrt n} \geq \sqrt {2 + \frac{40}{99}},$ which is better than the previous bounds $\sqrt 2$ (Shi, 1988), $\sqrt {2 + \frac{7654}{19071}}$ (Lai, 2017). We show that $\liminf_{n \rightarrow \infty} {g(n,m)-n\over \sqrt \frac{n}{m}} > \sqrt {2.444},$ for all even integers $m$. We make the following conjecture: $\liminf_{n \to \infty} {f(n)-n \over \sqrt n} > \sqrt {2.444}.$