Orientable vertex imprimitive complete maps

Xue Yu; Xue Yu

doi:10.3934/era.2024113

Electronic Research Archive

2024, Volume 32, Issue 4: 2466-2477. doi: 10.3934/era.2024113

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Research article

Orientable vertex imprimitive complete maps

Xue Yu ^,

Department of Mathematics, Henan Institute of Science and Technology, Xinxiang 453003, China

Received: 22 January 2024 Revised: 03 March 2024 Accepted: 14 March 2024 Published: 26 March 2024

In the work by Li (J. Combin. Theory Ser. B, 99 (2009), 447–454.), the author characterized the classification of vertex transitive embeddings of complete graphs, and proposed how to enumerate such maps. In this paper, we study the counting problem of orientable vertex imprimitive complete maps, which is the automorphism group of this map acts imprimitively on its vertex set. Moreover, we obtain the number of non-isomorphic embeddings when the vertex-stabilizer subgroups of the automorphism groups of maps are isomorphic to $ \text{Z}_{p-1} $ with odd prime $ p $.
- vertex imprimitive embeddings,
- complete graphs,
- Frobenius groups
Citation: Xue Yu. Orientable vertex imprimitive complete maps[J]. Electronic Research Archive, 2024, 32(4): 2466-2477. doi: 10.3934/era.2024113

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Abstract

In the work by Li (J. Combin. Theory Ser. B, 99 (2009), 447–454.), the author characterized the classification of vertex transitive embeddings of complete graphs, and proposed how to enumerate such maps. In this paper, we study the counting problem of orientable vertex imprimitive complete maps, which is the automorphism group of this map acts imprimitively on its vertex set. Moreover, we obtain the number of non-isomorphic embeddings when the vertex-stabilizer subgroups of the automorphism groups of maps are isomorphic to $ \text{Z}_{p-1} $ with odd prime $ p $.

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