Journal of Combinatorial Theory, Series B 80, 356370 (2000)
Face Size and the Maximum Genus of a Graph
1. Simple Graphs
Department of Mathematics, Normal University of Hunan,
Changsha 410081, People's Republic of China
Department of Mathematics, Northern Jiaotong University,
Beijing 100044, People's Republic of China
Received May 7, 1997
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. 519529, Physica
Verlag, Heidelberg). The other conjecture will be proved in a sequel to this paper.
2000 Academic Press
All graphs considered in this paper are finite and undirected and, unless
explicitly stated otherwise, they are also connected. In general, we allow
graphs to have loops and multiple edges. Graphs which lack both loops
and multiple edges will be called simple.
By a surface S we mean a compact connected 2-dimensional manifold
without boundary (that is, a closed surface). We consider both orientable
and nonorientable surfaces. It s well known that each orientable surface is
homeomorphic to a sphere with h handles while every nonorientable one
is homeomorphic to a sphere with k crosscaps. This number h or k is called
the genus, denotes by g(S), of the surface S when S is orientable or non-
orientable, respectively. A cellular embedding j: G Ä S of a graph G on a
doi:10.1006Âjctb.2000.1990, available online at http:ÂÂwww.idealibrary.com on
Copyright 2000 by Academic Press
All rights of reproduction in any form reserved.
Project supported by National Science Foundation of China.