On the Connectivity of Graphs Embedded in Surfaces

Abstract

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 γ >0 (respectively, nonorientable genus γ >0, γ ≠2) there is a complete graph of orientable genus γ (respectively, nonorientable genus γ ) and having connectivity attaining his bound. It is false that there is a complete graph of genus γ (respectively, nonorientable genus γ ), for every γ (respectively γ ) and that is the starting point of the present paper. Ringel and Youngs did show that for each γ >0 (respectively, γ >0, γ ≠2) there is a complete graph K n which embeds in S γ (respectively N γ ) such that n is the chromatic number of surface S γ (respectively, the chromatic number of surface N γ ). One then easily observes that the connectivity of this K n 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 κ max of the orientable surface S γ to be the maximum connectivity of any graph embeddable in the surface and the genus connectivity κ gen ( S γ ) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the bounds κ max ( N γ ) and κ gen ( N γ ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivity κ max ( S γ ) or κ max ( N γ ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [ γ ( K n )+1, γ ( K n +1 )] and [ γ ( K n )+1, γ ( K n +1 )] with respect to their genus and maximum connectivities.