On minimum-genus embeddings

Abstract

The cycle double cover conjecture is equivalent to the ‘pseudosurface embedding conjecture’ that every 2-connected graph has a closed 2-cell embedding in some pseudosurface. The ‘strong embedding conjecture’ asserts that every 2-connected graph has a closed 2-cell embedding in some surface. The concern of this paper is an even stronger topological conjecture mentioned by Seymour-the ‘genus strong embedding conjecture’, that every bridgeless cubic graph has a closed 2-cell embedding in its minimum genus surface. A surface Σ is said to have the genus strong embedding property if every 2-connected graph for which it is the minimum-genus surface has a closed 2-cell embedding in Σ. It is well-known that the sphere has the genus strong embedding property. Negami, and Robertson and Vitray have proved that the projective plane also has the genus strong embedding property. We prove in this paper a structure theorem for minimum-genus embeddings and embeddings with the minimum number of repeated vertices and edges in their facial walks (if the strong embedding conjecture is true, then this number is zero). This structure property leads to upper bounds on the number of repeated vertices and edges in the facial walks for such embeddings of 3-connected graphs. Examples are given that show these bounds are the best possible for minimum-genus embeddings. These examples also show that the sphere and the projective plane are the only surfaces having the genus strong embedding property, thereby extending Xuong's counterexample graph for the torus. An open problem mentioned in Bender and Richmond's paper 1990 is also solved.