This paper shows that a simple graph which can be cellularly embedded on some closed surface in such a way that the size of each face does not exceed 7 is upper embeddable. This settles one of two conjectures posed by Nedela and S8 koviera (1990, in "Topics in Combinatorics and Graph Theory," pp. 519-529, Physica Verlag, Heidelberg). The other conjecture will be proved in a sequel to this paper.

结合边连通性，本文给出了一个图的Betti亏数由这个图的补图的着色数所确定的上界式，证明了所给出的上界式是最好的，得到关于图的最大亏格下界的若干新结果。

设G为连通图且L是G的一条双向2-重迹。作者引入G的一个新参数，称之为G的反射数，并用ε（G）表示。反射数ε（G）由如下式子给出：ε（G）=minε（G，L），这里ε（G。L）是G的关于L。的反射数，且“min””取遍G的所有双向2-重迹L。然后，对于3-正则图G，作者证明了G的反射数ε（G）与G的最人亏格γM（G）密切相关，具体地，ε（G）=2γM,（G）-β（G），其中β（G）是G的圈秩数。同时，作者给出一个与ε（G）的值有关的G的特征结构。这些可视为ThomassenC的有关结果的进一步补充。

It is proved that a loopless graph which can be cellularly embedded on some closed surface in such a way that the size of each face does not exceed is upper embeddable. This settles the rst of two conjectures posed by Nedela and Skoviera in [NEDELA, R.-SKOVIERA, M.: On graphs beddable with short faces. In: Topics in Combinatorics and Graph Theory (R. Bodendiek, R. Henn, eds.), Physica Verlag, Heidelberg, 1990, pp. 519-529]. The second conjecture is established in [HUANG, Y.-LIU, Y.: Face size and the maximum genus of a graph. Part 1: Simple graphs, J. Combin. Theory Ser. B 80 (2000), 356-370].