On the Connectivity of Graphs Embedded in Surfaces

In a 1973 paper, Cooke obtained an upper bound on the possible connectivity of a graph embedded in a surface (orientable or nonorientable) of fixed genus. Furthermore, he claimed that for each orientable genus #>0 (respectively, nonorientable genus # >0, # {2) there is a complete graph of orientable genus # (respectively, nonorientable genus # ) and having connectivity attaining his bound. It is false that there is a complete graph of genus # (respectively, nonorientable genus # ), for every # (respectively # ) and that is the starting point of the present paper. Ringel and Youngs did show that for each #>0 (respectively, # >0, # {2) there is a complete graph Kn which embeds in S# (respectively N# ) such that n is the chromatic number of surface S# (respectively, the chromatic number of surface N# ). One then easily observes that the connectivity of this Kn attains the upper bound found by Cook. This leads us to define two kinds of connectivity bound for each orientable (or nonorientable) surface. We define the maximum connectivity }max of the orientable surface S# to be the maximum connectivity of any graph embeddable in the surface and the genus connectivity }gen(S#) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the bounds }max(N# ) and }gen(N# ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivity }max(S#) or }max(N# ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [#(Kn)+1, #(Kn+1)] and [# (Kn)+1, # (Kn+1)] with respect to their genus and maximum connectivities. 1998 Academic Press Article No. TB971809