TITLE: Graph chromatic number and graph homomorphism?
QUESTION [6 upvotes]: For two graphs G and H such that $\chi(G) < \chi(H)$, then is it true that there always exist a graph homomorphism from G to H ?

REPLY [3 votes]: No. Let $G$ be a triangle and $H$ the Grötzsch graph.
Then $\chi(G)=3$, $\chi(H)=4$ and since $H$ has no triangle, there is no homomorphism from $G$ to $H$.