Abstract: Let $K_{m,m,m}$ ($m\geq1$) be a complete regular tripartite graph, and $G$ be a bipartite graph with girth greater than 4. In this paper, the genus of cartesian product of $K_{m,m,m}$ and $G$ with $\Delta(G)\leq 2m$ is determined. It generalizes the result by Bonnington and Pisanski, which gives the genus of cartesian product of $K_{m,m,m}$ and an even cycle. Moreover, the nonorientable genera of cartesian products of $K_{m,m,m}$ and some non-bipartite graphs are obtained.

Key words: Genu, Complete regular tripartite graph, Bipartite graph, Cartesian product