Pentagons in triangle-free graphs

Abstract

For all $n\ge 9$, we show that the only triangle-free graphs on $n$ vertices maximizing the number $5$-cycles are balanced blow-ups of a 5-cycle. This completely resolves a conjecture by Erdős, and extends results by Grzesik and Hatami, Hladký, Král', Norin and Razborov, where they independently showed this same result for large $n$ and for all $n$ divisible by $5$.

6 pages

Cited by in corpus (6)

• Inducibility of directed paths
• Maximizing five-cycles in $K_r$-free graphs
• Semidefinite Programming and Ramsey Numbers
• $C_5$ is almost a fractalizer
• Minimizing the number of 5-cycles in graphs with given edge-density
• Inducibility in $H$-free graphs and inducibility of Turán graphs