An induced cycle is a cycle that is an induced subgraph of G; induced cycles are also called chordless cycles or (when the length of the cycle is four or more) holes.


Can anyone please tell me what is the maximum number of holes that a simple graph on n vertices can have? 


How to prove a polynomial upper-bound (i.e. $O(n^k)$ for some fixed $k$) for that? 



Help on the following extremal problem?
If you devide your set of vertices into $k$ ($k \in \mathbb{Z}\_{\geq 3}$) sets $V\_1, V\_2,\dots,V\_k$ and take all edges from $V\_i$ to $V\_{i+1\ (mod\ k)}$, then you get $(n/k)^k$ holes of length $k$. This is an exponential nunber so there is no polynomial upper bound. I dont know what the precise maximum is. 


