2-semi equivelar maps on the torus and the Klein bottle with few vertices

Document Type : Original paper


1 Department of Applied Science, Indian Institute of Information Technology, Allahabad 211 015, India

2 Department of Mathematics and Statistics, Vignan's Foundation for Science, Technology & Research, Vadlamudi 522213, India

3 Department of Applied Science & Humanities, Rajkiya Engineering College, Banda 210201, India


The $k$-semi equivelar maps, for $k \geq 2$, are generalizations of maps on the surfaces of Johnson solids to closed surfaces other than the 2-sphere. In the present study, we determine 2-semi equivelar maps of curvature 0 exhaustively on the torus and the Klein bottle. Furthermore, we classify (up to isomorphism) all these 2-semi equivelar maps on the surfaces with up to 12 vertices.


Main Subjects

[1] U. Brehm, Polyhedral Maps with Few Edges, Topics in Combinatorics and Graph Theory: Essays in Honour of Gerhard Ringel, Springer, 1990, pp. 153–162.
[2] U. Brehm and W. K¨uhnel, Equivelar maps on the torus, Eur. J. Comb. 29 (2008), no. 8, 1843–1861.
[3] D. Chavey, Tilings by regular polygons—II, Comput. Math. Appl. 17 (1989), no. 1–3, 147–165.
[4] B. Datta and D. Maity, Semi-equivelar and vertex-transitive maps on the torus, Beitr. Algebra Geom. 58 (2017), no. 3, 617–634.
[5] B. Datta and D. Maity, Platonic solids, archimedean solids and semi-equivelar maps on the sphere, Discrete Math. 345 (2022), no. 1, Article ID: 112652.
[6] B. Datta and N. Nilakantan, Equivelar polyhedra with few vertices, Discrete Comput. Geom. 26 (2001), no. 3, 429–461.
[7] B. Datta and A.K. Upadhyay, Degree-regular triangulations of torus and klein bottle, Proc. Math. Sci. 115 (2005), no. 3, 279–307.
[8] B. Grünbaum and G.C. Shephard, Tilings and Patterns, Courier Dover Publications, New York, 1987.
[9] J. Karabáš and R. Nedela, Archimedean solids of genus two, Electron. Notes Discrete Math. 28 (2007), 331–339.
[10] J. Karabáš and R. Nedela, Archimedean maps of higher genera, Math. Comp. 81 (2012), 569–583.
[11] W. Kurth, Enumeration of platonic maps on the torus, Discrete Math. 61 (1986), no. 1, 71–83.
[12] T. Réti, E. Bitay, and Z. Kosztol´anyi, On the polyhedral graphs with positive combinatorial curvature, Acta Polytech. Hung. 2 (2005), no. 2, 19–37.
[13] Y. Singh and A.K. Tiwari, Doubly semi-equivelar maps on the plane and the torus, AKCE Int. J. Graphs Comb. 19 (2022), no. 3, 296–310.
[14] Y. Singh and A.K. Tiwari, Enumeration of doubly semi-equivelar maps on the klein bottle, Indian J. Pure Appl. Math. (2023), 1–24.
[15] A.K. Tiwari, Y. Singh, and A. Tripathi, 2-semi-equivelar maps on the torus and the klein bottle with few vertices, (2022).
[16] A.K. Tiwari and A.K. Upadhyay, Semi-equivelar maps on the torus and the klein bottle with few vertices, Math. Slovaca. 67 (2017), no. 2, 519–532.
[17] A.K. Tiwari and A.K. Upadhyay, Semi-equivelar maps on the surface of euler characteristic−1, Note Mat. 37 (2018), no. 2, 91–102.
[18] A.K. Upadhyay, A.K. Tiwari, and D. Maity, Semi-equivelar maps, Beitr. Algebra Geom. 55 (2014), no. 1, 229–242.
[19] L. Zhang, A result on combinatorial curvature for embedded graphs on a surface, Discrete Math. 308 (2008), no. 24, 6588–6595.