# Maximising the Number of Cycles in Graphs with Forbidden Subgraphs

## TL;DR

This paper proves that for large graphs avoiding a certain subgraph H, the maximum number of cycles is achieved by a specific Turán graph, resolving longstanding questions in extremal graph theory.

## Contribution

It establishes the extremal structure for maximizing cycles in H-free graphs with chromatic number k+1, confirming a conjecture and answering a key question.

## Key findings

- The maximum number of cycles in large H-free graphs is achieved by the Turán graph T_k(n).
- The result applies to graphs with forbidden subgraphs H having chromatic number k+1 and a critical edge.
- The paper resolves a conjecture and a question posed by Arman, Gunderson, and Tsaturian.

## Abstract

Fix $k \ge 2$ and let $H$ be a graph with $\chi(H) = k+1$ containing a critical edge. We show that for sufficiently large $n$, the unique $n$-vertex $H$-free graph containing the maximum number of cycles is $T_k(n)$. This resolves both a question and a conjecture of Arman, Gunderson and Tsaturian.