  We give an efficient randomized algorithm to construct a box representation
of any graph G on n vertices in $1.5 (\Delta + 2) \ln n$ dimensions, where
$\Delta$ is the maximum degree of G. We also show that $\boxi(G) \le (\Delta +
2) \ln n$ for any graph G. Our bound is tight up to a factor of $\ln n$. We
also show that our randomized algorithm can be derandomized to get a polynomial
time deterministic algorithm. Though our general upper bound is in terms of
maximum degree $\Delta$, we show that for almost all graphs on n vertices, its
boxicity is upper bound by $c\cdot(d_{av} + 1) \ln n$ where d_{av} is the
average degree and c is a small constant. Also, we show that for any graph G,
$\boxi(G) \le \sqrt{8 n d_{av} \ln n}$, which is tight up to a factor of $b
\sqrt{\ln n}$ for a constant b.
