Maximising the number of induced cycles in a graph

TL;DR

This paper determines the maximum number of induced cycles in large graphs, identifies the unique extremal graph, and resolves longstanding conjectures about cycle counts and extremal structures.

Contribution

It establishes the maximum counts of induced, odd, and even cycles in large graphs and characterizes the unique extremal graphs achieving these maxima.

Findings

Maximum number of induced cycles in large graphs identified

Unique extremal graph for induced cycles characterized

Resolved conjectures on cycle counts from 1988

Abstract

We determine the maximum number of induced cycles that can be contained in a graph on $n\ge n_0$ vertices, and show that there is a unique graph that achieves this maximum. This answers a question of Tuza. We also determine the maximum number of odd or even cycles that can be contained in a graph on $n\ge n_0$ vertices and characterise the extremal graphs. This resolves a conjecture of Chv\'atal and Tuza from 1988.