Hamilton surfaces for the complete even symmetric bipartite graph (Q1263593)

A hamilton surface for a connected graph G is defined as an imbedding of G into a surface so that all faces of the imbedding are r-gons for some fixed \(r\geq 3\). Then a collection of hamilton surfaces for G such that every r-cycle in G is the boundary of a face is precisely one surface is called a hamilton surface decomposition of G. The main result of this note is that \(K_{2n,2n}\) has an orientable hamilton surface decomposition, for \(r=4\), into representative classes \(A_{1,1},A_{1,2},...,A_{2n-1,2n-1}\), such that \(A_{i,j}\) and \(A_{k,\ell}\) form a hamilton surface whenever \((i-k,2n-1)=1\) and \((j- 1),2n-1)=1\). (Each representative class is a set of squares in \(K_{2n,2n}\) containing every edge of \(K_{2n,2n}\) exactly once.)