  In this short note, we relate the boxicity of graphs (and the dimension of
posets) with their generalized coloring parameters. In particular, together
with known estimates, our results imply that any graph with no $K_t$-minor can
be represented as the intersection of $O(t^2\log t)$ interval graphs (improving
the previous bound of $O(t^4)$), and as the intersection of $\tfrac{15}2 t^2$
circular-arc graphs.
