# The maximum number of cycles in a graph with fixed number of edges

## TL;DR

This paper investigates the maximum number of cycles in graphs with fixed edges or vertices, providing new bounds and examples that improve upon previous conjectures and results.

## Contribution

It establishes new upper bounds on the number of cycles in graphs with fixed edges, and presents examples demonstrating these bounds are tight for large graphs.

## Key findings

- Graphs with m edges have at most (1.443)^m cycles for large m.
- Examples exist with (1.37)^m cycles, showing bounds are nearly tight.
- New bounds are also provided for multigraphs with fixed edges or vertices.

## Abstract

The main topic considered is maximizing the number of cycles in a graph with given number of edges. In 2009, Kir\'aly conjectured that there is constant $c$ such that any graph with $m$ edges has at most $(1.4)^m$ cycles. In this paper, it is shown that for sufficiently large $m$, a graph with $m$ edges has at most $(1.443)^m$ cycles. For sufficiently large $m$, examples of a graph with $m$ edges and $(1.37)^m$ cycles are presented. For a graph with given number of vertices and edges an upper bound on the maximal number of cycles is given. Also, exponentially tight bounds are proved for the maximum number of cycles in a multigraph with given number of edges, as well as in a multigraph with given number of vertices and edges.