Bi-embeddings of graphs

Abstract

Let γ and γ' be non-negative integers. We say that the graph G is (γ, γ') bi-embeddable if G can be embedded in a surface of genus γ and the complement Ḡ of G can be embedded in a surface of genus γ'. Let N(γ, γ') be the least integer such that every graph with at least N(γ, γ') points is not (γ, γ') bi-embeddable. It has been shown in [1] and [5] that N(0, 0) = 9; this result was also obtained by John R. Ball of the Carnegie Institute of Technology. Our object here is to obtain upper and lower bounds for N(γ, γ').