For any graph $G$ let $\eta(G)$ be the Hadwiger number of $G$.
Is there for every graph $G$ a graph $2G$ such that
-- $\chi(2G) = 2\chi(G)$, and
-- $\eta(2G) = 2\eta(G)$?
For each one of the above conditions it is easy to construct a graph $2G$ to $G$ such that the condition holds, but I haven't been able to double both the coloring number and the Hadwiger number at the same time with a "universal construction".