Let $N>2$ be a positive integer and $G$ be a simple graph satisfies:
- the maximal degree of $G$ is $N$
- the clique number of $G$ is $N$.
I want to ask if there exists a vertex independent set $I$ in $V(G)$ such that for every $N$-order complete subgraph $H$ of $G$, the intersection of $I$ and $V(H)$ is not empty. If not, please give a counterexample.