In the book embedding of a graph $G$ , each vertex of $G$ is placed on the spine and each edge is placed in the pages without crossing each other edge. If vertices have degree at most one in each page, the book embedding is matching . The minimum number of pages in which a graph can be matching book embedded is called matching book thickness. For Convenience, we denote the matching book thickness of a graph $G$ by $\mathrm{mbt}(G)$.
For the Cartesian product of a complete graph $K_n$ and a path $P_m$, I want to know $\mathrm{mbt}(K_n\Box P_m)$. Since $\mathrm{mbt}(K_n)=n$, it is not hard to see that $\mathrm{mbt}(K_n\Box P_2)=n+1,$ where $P_2=K_2$. For the case $ K_n\Box P_3$, I guess $\mathrm{mbt}(K_n\Box P_3)=n+2$. But I have no idea about the proof.
I will appreciate it if someone could give any suggestions.