TITLE: Homomorphism for a fixed graph NP-complete?
QUESTION [9 upvotes]: Let $G$ be the following Graph:

We want to decide whether for an input structure $\mathcal{S}$ there exists a homomorphism $S \to G$. We will call this problem $HOM_G$.
The task at hand is to show that $HOM_G$ is NP-complete. This is a bit strange since $HOM_G$ is not a general problem but a very specific one.
I've already made some observations: $HOM_G \in NP$ since if a potential homomorphism is given we can decide in polynomial time if it is correct. Furthermore all positive instances $\mathcal{S}$ have to be graphs because if there is a non-empty non-binary or more then one binary relation there is no homomorphism to $G$.
I'm a little bit stuck right now. Any ideas?

REPLY [3 votes]: By reduction from 3-colorability.
This graph injects in your $G$ in exactly three different ways:

If you have a graph that you want to 3-color, create one copy of the above hexagon for each of its nodes, and represent each of its edges by two successive edges -- e.g. like this if the original graph is a square with a diagonal:

The direction of the links that represent the original edges doesn't matter -- either way it allows neighboring hexagons to embed into $G$ in any two different ways, but not in the same way.
Thus there is a homomorphism from the expanded graph to $G$ exactly if the original graph was 3-colorable.
(Partial idea credit to @domotorp's now deleted answer).