  Let $G$ be a graph of order $n$. A path decomposition $\mathcal{P}$ of $G$ is
a collection of edge-disjoint paths that covers all the edges of $G$. Let
$p(G)$ denote the minimum number of paths needed in a path decomposition of
$G$. Gallai conjectured that if $G$ is connected, then $p(G)\leq
\lceil\frac{n}{2}\rceil$. In this paper, we prove that Gallai's path
decomposition conjecture holds for the cartesian product $G\Box H$, where $H$
is any graph and $G$ is a unicyclic graph or a bicyclic graph.
