If $G= (V(G), E(G))$ and $H=(V(H), E(H))$ are graphs.

Consider the set $\mathfrak{R}(G,H)$ of "Random Cartesian Product" whose member are graphs $K =(V(K), E(K))$ defined as follow:

$V(K)$ = $V(G)$ x $V(H)$ and

If $uv \in E(G)$ and $xy \in E(H)$ then either:

{$(u,x)(v, x), (u,y)(v,y)$} $\subset$ $E(K)$

or

{$(u,x)(u,y), (v,x)(v,y)$} $\subset$ $E(K)$

But not both.

1- Is it true that if $K \in \mathfrak{R}(G,H)$, then $K$ either contains a copy of $G$ or a copy of $H$?

2- If $\chi(G)$, $\chi(H)$ $\gt k$, what could be said about $\chi(K)$ when $K \in \mathfrak{R}(G,H)$?