中文摘要：

图在曲面上的可嵌入性是拓扑图论的主要问题之一.在刘彦佩提出的联树模型的基础上,通过一个图在曲面上的嵌入可用其联树,进一步其关联曲面来表示,然后逐层分段,得到了完全二部图km,n,至少有C1C2m/2C3m/2C4mn（m-C5）-n（m-C6）mn/2（m-1）m-1/2（n-1）n-1/2个不同的最小亏格嵌入,其中常量C1,C2,C3,C4,C5和C6依赖于m模和n模4的余数.此结论改进了文献[8]中结果.

英文摘要：

The embeddability of a graph ona surface is one of major problems in topological graph theory. Based on the joint trees, an embedding of a graph on a surface can be represented by a joint tree, further by an associated surface of it. By dividing the associated surfaces into segments layer by layer, the number of genus embeddings of a complete bipartite m n mn 1 I graph Km.n is derived, namely C1C2m/2C3m/2C4mn（m-C5）-n（m-C6）mn/2（m-1）m-1/2（n-1）n-1, where C1, C2, C3, Ca, C5 and C6 are constants depending on the residual class ofm modular 4 and that ofn modular 4.