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Regular and orientably-regular maps with quasiprimitive automorphism groups on vertices

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Abstract

Regular and orientably-regular maps are central to the part of topological graph theory concerned with highly symmetric graph embeddings. Classification of such maps often relies on factoring out a normal subgroup of automorphisms acting intransitively on the set of the vertices of the map. Maps whose automorphism groups act quasiprimitively on their vertices do not allow for such factorization. Instead, we rely on classification of quasiprimitive group actions which divides such actions into eight types, and we show that four of these types, HS, HC, SD, and CD, do not occur as the automorphism groups of regular or orientably-regular maps. We classify regular and orientably-regular maps with automorphism groups of the HA type, and construct new families of regular as well as both chiral and reflexible orientably-regular maps with automorphism groups of the TW and PA types. We provide a brief summary of the known results concerning the AS type, which has been extensively studied before.

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Acknowledgements

Yan Wang was introduced to the topic of this paper during her research visit with Professor Širáň in Bratislava, Slovakia, from November 2015 to May 2016. She is very thankful for the hospitality she experienced during her visit; both at the Slovak University of Technology as well as at the Comenius University. In particular, she wishes to express her gratitude toward Professor Širáň who proved to be a wonderful host, guide, and colleague. She also wishes to thank Professor Haipeng Qu of Shanxi Normal University, China, for his useful advise concerning the structure of quasiprimitive groups. The authors are very thankful to the anonymous referee for all her/his suggestions for improvements.

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Authors and Affiliations

  1. Comenius University, Bratislava, Slovakia

    Robert Jajcay

  2. Department of Mathematics, Southern University of Science and Technology, Shenzhen, 518055, Guangdong, People’s Republic of China

    Cai-Heng Li

  3. Slovak University of Technology, Bratislava, Slovakia

    Jozef Širáň

  4. Yan Tai University, Yantai, People’s Republic of China

    Yan Wang

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  1. Robert Jajcay
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  2. Cai-Heng Li
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  3. Jozef Širáň
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  4. Yan Wang
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Correspondence to Robert Jajcay.

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Yan Wang: Supported by NSFC 11371307, 11671347, 61771019, NSFS Nos. ZR2017 MA022 and J16LI02. Cai-Heng Li: Supported by NSFC 11231008, partially supported by NSFC grant 11771200. Jozef Širáň: Supported by APVV 0136/12, APVV-15-0220, VEGA 1/0026/16, VEGA 1/0142/17 and NSFC 11371307. Robert Jajcay: Supported by VEGA 1/0596/17, VEGA 1/0719/18, APVV-15-0220, by the Slovenian Research Agency (Research Projects N1-0038, N1-0062, J1-9108), and by NSFC 11371307.

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Jajcay, R., Li, CH., Širáň, J. et al. Regular and orientably-regular maps with quasiprimitive automorphism groups on vertices. Geom Dedicata 203, 389–418 (2019). https://doi.org/10.1007/s10711-019-00440-6

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  • DOI: https://doi.org/10.1007/s10711-019-00440-6

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