Maximum density of an induced 5-cycle is achieved by an iterated blow-up of a 5-cycle

TL;DR

The paper determines the maximum number of induced 5-cycles in large graphs and shows that this maximum is achieved by a specific iterated blow-up construction of a 5-cycle.

Contribution

It establishes the exact maximum count of induced 5-cycles and characterizes the extremal graphs as iterated blow-ups of a 5-cycle for large n.

Findings

Maximum number of induced 5-cycles is given by a specific formula involving partitioning of vertices.

For n a power of 5, the extremal graph is uniquely an iterated blow-up of a 5-cycle.

The extremal construction asymptotically maximizes induced 5-cycles in large graphs.

Abstract

Let $C(n)$ denote the maximum number of induced copies of 5-cycles in graphs on $n$ vertices. For $n$ large enough, we show that $C(n)=a\cdot b\cdot c \cdot d \cdot e + C(a)+C(b)+C(c)+C(d)+C(e)$, where $a+b+c+d+e = n$ and $a,b,c,d,e$ are as equal as possible. Moreover, if $n$ is a power of 5, we show that the unique graph on $n$ vertices maximizing the number of induced 5-cycles is an iterated blow-up of a 5-cycle.