Gallai's path decomposition conjecture for Cartesian product of graphs

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 $p(G)\le \lceil \frac{n}{2}\rceil$. Let $n_o(G)$ to denote the number of vertices with odd degree in $G$. Lovász proved that if $G$ is a connected graph with all vertices having degree odd, i.e. $n_o(G)=n$, then $p(G)=\frac{n}{2}$. In this paper, we prove that if $G$ is a connected graph of order $m\ge 2$ with $p(G)=\frac{n_o(G)}{2}$ and $H$ is a connected graph of order $n$, then $p(G\mathrm{\square }H)\le \frac{mn}{2}$. Furthermore, we prove that $p(G)=\frac{n_o(G)}{2}$, if one of the following is hold: (\romannumeral1) $G$ is a tree; (\romannumeral2) $G=P_n\mathrm{\square }T$, where $n\ge 4$ and $T$ is a tree; (\romannumeral3) $G=P_n\mathrm{\square }H$, where $H$ is an even graph.