Let the category $\mathbf{Gr}$ consist of simple, undirected graphs, together with graph homomorphisms. We say that a graph $P$ is *projective* if for all graphs $A, B$ with a surjective graph homomorphism $s:A\to B$ and every graph homomorphism $h:P\to B$ there is a graph homomorphism $f:P\to A$ such that $$h = s\circ f.$$

As an example, for $n>2$ the complete graph $K_n$ is not projective: Let $B = K_n$, and let $A$ be a graph with $\chi(A) = n$ but $A$ contains no clique of size $n$. Let $s:A\to B$ be any coloring, let $h:K_n \to B$ be the identity, then there is no $f: K_n \to A$ such that $h=s\circ f$.

**Question.** Given an integer $n>2$, is there a projective graph $G$ with $\chi(G) =n$?