Abstract
Semi-Equivelar maps are generalizations of maps on the surface of Archimedean Solids to surfaces other than \(2\)-Sphere. We classify some semi-equivelar maps on surface of Euler characteristic \(-1\) and show that none of these are vertex transitive. We establish existence of \(12\)-covered triangulations for this surface. We further construct double cover of these maps to show existence of semi-equivelar maps on the surface of double torus. We also construct several semi-equivelar maps on the surfaces of Euler characteristics \(-8\) and \(-10\) and on non-orientable surface of Euler characteristics \(-2\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Altshuler, A., Brehm, U.: The weakly neighbourly polyhedral maps on the 2-manifold with Euler characteristic \(-\)1. Discrete Comput. Geom. 1, 355–369 (1986)
Babai, L.: Vertex-transitive graphs and vertex-transitive maps. J. Graph Th. 15(6), 587–627 (1991)
Brehm, U., Kühnel, W.: Equivelar maps on torus. Eur. J. Comb. 29, 1843–1861 (2008)
Brehm, U., Schulte, E.: Polyhedral maps. In: Handbook of discrete and computational geometry, pp. 345–358. CRC Press Ser. Discrete Math. Appl, CRC, Boca Raton (1997)
Conder Marston D.E., Dobcsanyi, P.: Determination of all regular maps of small genus, J. Comb. Theory Ser. B 81:224–242, (2001)
Conder Marston D.E.: Regular maps and hypermaps of Euler characteristic \(-1\) to \(-200\). J. Comb. Theory Ser. B, 99:455–459 (2009)
Coxeter H. S. M., Moser, W. O. J.: Generators and relations for discrete groups, 4th edn. Springer, Heidelberg, (1980)
Datta, B.: A note on the existence of \(\{k, k\}\)-equivelar polyhedral maps. Beiträge zur Algebra und Geometrie 46, 537–544 (2005)
Datta, B., Upadhyay, A. K.: Degree regular triangulations of torus and Klein bottle. Proc. Indian Acad. Sci. (Math. Sc.), 115, (2005)
Datta, B. Upadhyay, A. K.: Degree regular triangulations of double torus, Forum Mathematicum (2006)
Karabas, J., Nedela, R.: Archimedean solids of genus two. Electron Notes Discrete Math. 28, 331–339 (2007)
Karabas, J., Nedela, R.: Archimedean maps of higher genera. Math. Comput. 81, 569–583 (2012)
Program to Compute Reduced Homology Groups hom gap. (2012)
The GAP Group: GAP-Groups, Algorithms, and Programming, Version 4.4.12; 2008 (http://www.gapsystem.org)
Lutz, F., Sulanke, T., Tiwari, A. K., Upadhyay, A. K.: Equivelar and d-covered triangulations on surfaces-I http://arxiv.org/abs/1001.2777 (2010)
Lutz, F.: http://page.math.tu-berlin.de/lutz/stellar/vertex-transitive-triangulations.html (2012)
McMullen, P., Schulz, Ch., Wills, J.M.: Polyhedral 2-manifolds in \(E^3\) with unusually large genus. Israel J. Math. 46, 127–144 (1983)
Negami, S., Nakamoto, A.: Triangulations on closed surfaces covered by vertices of given degree. Graphs Comb. 17, 529–537 (2001)
Pellicer, D.: Vertex-transitive maps with Schläfli type \(\{3, 7\}\). http://arxiv.org/abs/1110.5977v1 (2012)
Pellicer, D., Ivic Weiss, A.: Unifrom maps on surfaces of non-negative Euler characteristic. Symmetry Cult Sci (special issue on Tesselations) 22, 159–196 (2011)
Upadhyay, A.K.: A note on Upper Bound for d-covered Triangulations of Closed Surfaces. Int. J. Pure App. Math. 67, 1–5 (2011)
Upadhyay, A. K., Tiwari, A. K.: Semi Equivelar Maps on Torus and Klein bottle (2012)
Upadhyay, A. K., Tiwari, A. K., Maity, D.: Some semi-equivelar maps. http://arxiv.org/abs/1101.0671 (2012)
Acknowledgments
Part of this work was done when the first author was visiting Department of Mathematics, Indian Institute of Science during June–July 2010. We would like to thank Prof. B. Datta for numerous suggestions which led to significant improvements in the article. We would also like to thank Prof. S. C. Gupta whose suggestions proved valuable. The authors will also like to express their gratitude to anonymous referee whose pointed out many corrections and whose suggestions have led to substantial improvement in this article. Work of first author is supported by DST research grant No. SR/S4/MS:717/10 and that of second author is supported by CSIR award No. 09/1023(0003)/2010-EMR-I.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Upadhyay, A.K., Tiwari, A.K. & Maity, D. Semi-equivelar maps. Beitr Algebra Geom 55, 229–242 (2014). https://doi.org/10.1007/s13366-012-0130-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s13366-012-0130-6