On the Number of Pentagons in Triangle-Free Graphs

TL;DR

This paper proves the maximum number of pentagons in large triangle-free graphs using flag algebras, confirming a conjecture by Erdős and characterizing extremal graphs.

Contribution

It establishes the exact maximum number of pentagons in triangle-free graphs and characterizes extremal structures, settling a long-standing conjecture.

Findings

Maximum pentagons in triangle-free graphs is (n/5)^5.

Extremal graphs are balanced blow-ups of a pentagon.

Conjecture by Erdős (1984) is confirmed.

Abstract

Using the formalism of flag algebras, we prove that every triangle-free graph $G$ with $n$ vertices contains at most $(n/5)^5$ cycles of length five. Moreover, the equality is attained only when $n$ is divisible by five and $G$ is the balanced blow-up of the pentagon. We also compute the maximal number of pentagons and characterize extremal graphs in the non-divisible case provided $n$ is sufficiently large. This settles a conjecture made by Erd\H{o}s in 1984.