| A near-triangular embedding is an embedded graph into some surface whose all but one facial walks are $3$-gons. In this paper we show that if a graph $G$ is a triangulation of an orientable surface $S_h$, then $G$ has a near-triangular embedding into $S_k$ for $k=h,h+1,\cdots,\lfloor\frac{\beta(G)}{2}\rfloor$, where $\beta(G)$ is the Betti number of $G$. |
