On the genus of the tensor product of graphs where one factor is a regular graph

AbstractThe tensor product H⊗G where G is a 2k-regular graph can be regarded as a covering space of the permutation voltage graph H(2k) obtained from H. Assuming that H is suitably imbedded in some orientable surface by modifying the edges of H according to the configuration of G we get the permutation voltage graph H(2k) whose permutation derived graph is exactly H⊗G. This construction can also be extended to the tensor product H⊗G where G is a (2k + 1)-regular graph with 1-factor. Here we put the sufficient conditions on H and G so that the permutation derived imbedding obtained in this way is a minimal imbedding. We also give sample results — the genus of the tensor products H⊗K2n,2n and H⊗Qn are calculated for certain graphs H