  In this paper, we relate the seemingly unrelated concepts of treewidth and
boxicity. Our main result is that, for any graph G, boxicity(G) <= treewidth(G)
+ 2. We also show that this upper bound is (almost) tight. Our result leads to
various interesting consequences, like bounding the boxicity of many well known
graph classes, such as chordal graphs, circular arc graphs, AT-free graphs,
co--comparability graphs etc. All our bounds are shown to be tight up to small
constant factors. An algorithmic consequence of our result is a linear time
algorithm to construct a box representation for graphs of bounded treewidth in
a space of constant dimension. We also show many structural results as a
consequence. In particular, we show that, if the boxicity of a graph is b >= 3,
then there exists a simple cycle of length at least b-3 as well as an induced
cycle of length at least floor of (log(b-2) to the base Delta) + 2, where Delta
is its maximum degree. We also relate boxicity with the cardinality of minimum
vertex cover, minimum feedback vertex cover etc. Another structural consequence
is that, for any fixed planar graph H, there is a constant c(H) such that, if
boxicity(G) >= c(H) then H is a minor of G.
