Let $G$ and $H$ be two non-bipartite graphs. We know that, if $\exists$ homomorphism $\phi : G \rightarrow H$, then $\omega(G) \le \omega(H)$ where $\omega$ is clique number. 


$(1)$ Does the converse hold in general? 


$(2)$ Under what conditions does the converse hold?



graph homomorphism
You have your inequality backwards, I believe.


If $\omega(G) \le \omega(H)$, it does not follow in general that there is a homomorphism from $G\to H$. There are many triangle-free graphs with chromatic number greater than four, none of these will admit a homomorphism to $K\_4$.


