Hamilton surfaces for the complete even symmetric bipartite graph

A cycle in a graph G is called a hamilton cycle if it contains every vertex of G. A l-factor of a graph G is a subgraph H of G with the same vertex set as G, such that each vertex of H has degree one. Ringel [S] has generalized the idea of a hamilton cycle to two dimensions. He showed that if n is odd the set of squares in the n-dimensional cube Q,, can be partitioned into subsets such that each subset forms a connected polyhedron on an orientable surface of genus (n 4)2n-3 + 1. He called these subsets hamilton surfaces. These polyhedra are in fact genus embeddings of the graph consisting of the vertices and edges of Qm. A hamilton surface for a graph G is an embedding of G into a surface with the property that all faces of the embedding are r-gons for some fixed r 3 3. A hamilton surface decomposition of G is a collection of hamilton surfaces such that every r-cycle in G is the boundary of a face in precisely one surface. In this paper, we shall consider hamilton surfaces for K2n,2n and r = 4. These surfaces are genus embeddings of Kti,zn. For K 2n,2n the genus embedding consists of 2n2 squares and the orientable genus is p = (n 1)2. Let us return for a moment to one dimension. The following theorem is well known.