Face Size and the Maximum Genus of a Graph 1. Simple Graphs

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 Škoviera (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.