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Enumeration of doubly semi-equivelar maps on the Klein bottle

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Abstract

A vertex v in a map M has the face-sequence \((p_1^{n_1}. p_2^{n_2}. \ldots . p_k^{n_k})\), if consecutive \(n_i\) numbers of \(p_i\)-gons are incident at v in the given cyclic order for \(1 \le i \le k\). A map is called semi-equivelar if the face-sequence of each vertex is same throughout the map. A doubly semi-equivelar map is a generalization of semi-equivelar map which has precisely 2 distinct face-sequences. In this article, we determine all the types of doubly semi-equivelar maps of combinatorial curvature 0 on the Klein bottle. We present classification of doubly semi-equivelar maps on the Klein bottle and illustrate this classification for those doubly semi-equivelar maps which comprise of face-sequence pairs \(\{(3^6), (3^3.4^2)\}\) and \(\{(3^3.4^2), (4^4)\}\).

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Acknowledgements

The first author is thankful to the Ministry of Education, New Delhi, India for financial support. The second author expresses his thanks to IIIT Allahabad for providing the facility and resources to carry this research work.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Vignan’s Foundation for Science, Technology & Research, Vadlamudi, A.P., 522213, India

    Yogendra Singh

  2. Department of Applied Sciences, Indian Institute of Information Technology Allahabad, Prayagraj, U.P., 211015, India

    Anand Kumar Tiwari

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  1. Yogendra Singh
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  2. Anand Kumar Tiwari
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Correspondence to Yogendra Singh.

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Communicated by Shariefuddin Pirzada.

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Singh, Y., Tiwari, A.K. Enumeration of doubly semi-equivelar maps on the Klein bottle. Indian J Pure Appl Math (2023). https://doi.org/10.1007/s13226-023-00503-1

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  • DOI: https://doi.org/10.1007/s13226-023-00503-1

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