  The product dimension of a graph G is defined as the minimum natural number l
such that G is an induced subgraph of a direct product of l complete graphs. In
this paper we study the product dimension of forests, bounded treewidth graphs
and k-degenerate graphs. We show that every forest on n vertices has a product
dimension at most 1.441logn+3. This improves the best known upper bound of
3logn for the same due to Poljak and Pultr. The technique used in arriving at
the above bound is extended and combined with a result on existence of
orthogonal Latin squares to show that every graph on n vertices with a
treewidth at most t has a product dimension at most (t+2)(logn+1). We also show
that every k-degenerate graph on n vertices has a product dimension at most
\ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by
Eaton and Rodl.
