A quadrangular embedding of a graph in a surface Σ, also known as a quadrangulation of Σ, is a cellular embedding in which every face is bounded by a 4-cycle. A quadrangulation of Σ is minimal if there is no quadrangular embedding of a (simple) graph of smaller order in Σ. In this paper we determine $n(\mathrm{\Sigma })$, the order of a minimal quadrangulation of a surface Σ, for all surfaces, both orientable and nonorientable. Letting $S_0$ denote the sphere and $N_2$ the Klein bottle, we prove that $n(S_0)=4,n(N_2)=6$, and $n(\mathrm{\Sigma })=\lceil (5+\sqrt{25-16\chi (\mathrm{\Sigma })})/2\rceil$ for all other surfaces Σ, where $\chi (\mathrm{\Sigma })$ is the Euler characteristic. Our proofs use a ‘diagonal technique’, introduced by Hartsfield in 1994. We explain the general features of this method.

图形在曲面 Σ 中的四边形嵌入，也称为Σ 的四边形，是一种细胞嵌入，其中每个面都以 4 循环为界。如果 Σ 中没有较小阶的（简单）图的四边形嵌入，则Σ 的四边形是最小的。在本文中，我们确定$n(\mathrm{\Sigma })$，对于所有曲面，包括可定向和不可定向的曲面 Σ 的最小四边形的阶数。让${\mathrm{小号}}_0$表示球体和${ñ}_2$克莱因瓶，我们证明$n({\mathrm{小号}}_0)=4,n({ñ}_2)=6$， 和$n(\mathrm{\Sigma })=\lceil (5+\sqrt{25-16\chi (\mathrm{\Sigma })})/2\rceil$对于所有其他曲面 Σ，其中$\chi (\mathrm{\Sigma })$是欧拉特征。我们的证明使用了 Hartsfield 在 1994 年引入的“对角线技术”。我们解释了这种方法的一般特征。