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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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