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On the genus of the tensor product of graphs where one factor is a regular graph. (English) Zbl 0812.05019
Authors’ abstract: The tensor product $$H\otimes G$$ 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 $$H\otimes G$$. This construction can also be extended to the tensor product $$H\otimes G$$ 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 $$H\otimes K_{2n,2n}$$ and $$H\otimes Q_ n$$ are calculated for certain graphs $$H$$.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
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