Enumeration of doubly semi-equivelar maps on the Klein bottle

Yogendra Singh () and Anand Kumar Tiwari ()
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Yogendra Singh: Vignan’s Foundation for Science, Technology & Research
Anand Kumar Tiwari: Indian Institute of Information Technology Allahabad

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 2, 571-594

Abstract: Abstract A vertex v in a map M has the face-sequence $$(p_1^{n_1}. p_2^{n_2}. \ldots . p_k^{n_k})$$ ( p 1 n 1 . p 2 n 2 . … . p k n k ) , if consecutive $$n_i$$ n i numbers of $$p_i$$ p i -gons are incident at v in the given cyclic order for $$1 \le i \le k$$ 1 ≤ i ≤ 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 $$\{(3^6), (3^3.4^2)\}$$ { ( 3 6 ) , ( 3 3 . 4 2 ) } and $$\{(3^3.4^2), (4^4)\}$$ { ( 3 3 . 4 2 ) , ( 4 4 ) } .

Keywords: Doubly semi-equivelar maps; Face-sequence; Combinatorial curvature; Klein bottle; 52B70; 05C10; 05C38; 05C30 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00503-1

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