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 genusγ>0 (respectively, nonorientable genusγ>0,γ≠2) there is a complete graph of orientable genusγ(respectively, nonorientable genusγ) and having connectivity attaining his bound. It is false that there is a complete graph of genusγ(respectively, nonorientable genusγ), forevery γ(respectivelyγ) and that is the starting point of the present paper. Ringel and Youngsdidshow that for eachγ>0 (respectively,γ>0,γ≠2) there is a complete graphKnwhich embeds inSγ(respectivelyNγ) such that n is the chromatic number of surfaceSγ(respectively, the chromatic number of surfaceNγ). One then easily observes that the connectivity of thisKnattains the upper bound found by Cook. This leads us to definetwokinds of connectivity bound for each orientable (or nonorientable) surface. We define themaximum connectivity κmaxof the orientable surfaceSγto be the maximum connectivity of any graph embeddable in the surface and thegenus connectivity κgen(Sγ) of the surface to be the maximum connectivity of any graph which genus embeds in the surface. For nonorientable surfaces, the boundsκmax(Nγ) andκgen(Nγ) are defined similarly. In this paper we first study the uniqueness of graphs possessing connectivityκmax(Sγ) orκmax(Nγ). The remainder of the paper is devoted to the study of the spectrum of values of genera in the intervals [γ(Kn)+1, γ(Kn+1)] and [γ(Kn)+1, γ(Kn+1)] with respect to their genus and maximum connectivities.