Let $N>2$ be a positive integer and $G$ be a simple graph satisfies:


1. the maximal degree of $G$ is $N$
2. 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.



vertex independent set and the maximal clique
If $G$ is an odd cycle (as commented above), the answer is no. If it's an $(n+1)$-clique, it meets your degree condition but not your condition on the clique number. And if it's neither a clique nor an odd cycle, then by [Brooks' theorem](https://en.wikipedia.org/wiki/Brooks%27_theorem), it has an $n$-coloring, all of whose color classes are independent sets that meet every $n$-clique.


