Previous |  Up |  Next

Article

Title: On chirality groups and regular coverings of regular oriented hypermaps (English)
Author: Breda d'Azevedo, Antonio
Author: Rodrigues, Ilda Inácio
Author: Fernandes, Maria Elisa
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 61
Issue: 4
Year: 2011
Pages: 1037-1047
Summary lang: English
.
Category: math
.
Summary: We prove that if the Walsh bipartite map $\mathcal {M}=\mathcal {W}(\mathcal {H})$ of a regular oriented hypermap $\mathcal {H}$ is also orientably regular then both $\mathcal {M}$ and $\mathcal {H}$ have the same chirality group, the covering core of $\mathcal {M}$ (the smallest regular map covering $\mathcal {M}$) is the Walsh bipartite map of the covering core of $\mathcal {H}$ and the closure cover of $\mathcal {M}$ (the greatest regular map covered by $\mathcal {M}$) is the Walsh bipartite map of the closure cover of $\mathcal {H}$. We apply these results to the family of toroidal chiral hypermaps $(3,3,3)_{b,c}=\mathcal {W}^{-1}\{6,3\}_{b,c}$ induced by the family of toroidal bipartite maps $\{6,3\}_{b,c}$. (English)
Keyword: hypermap
Keyword: regular covering
Keyword: chirality group
Keyword: chirality index
Keyword: toroidal hypermaps
MSC: 05C10
MSC: 05C25
MSC: 20B25
MSC: 20F65
MSC: 51E30
MSC: 57M07
MSC: 57M60
idZBL: Zbl 1249.05086
idMR: MR2886255
DOI: 10.1007/s10587-011-0046-6
.
Date available: 2011-12-16T15:45:28Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/141805
.
Reference: [1] Breda, A., d'Azevedo, A. Breda, Nedela, R.: Chirality group and chirality index of Coxeter chiral maps.Ars Comb. 81 (2006), 147-160. MR 2267808
Reference: [2] d'Azevedo, A. Breda: A theory of restricted regularity of hypermaps.J. Korean Math. Soc. 43 (2006), 991-1018. MR 2303613, 10.4134/JKMS.2006.43.5.991
Reference: [3] d'Azevedo, A. Breda, Jones, G. A.: Double coverings and reflexive abelian hypermaps.Beitr. Algebra Geom. 41 (2000), 371-389. MR 1801428
Reference: [4] d'Azevedo, A. Breda, Jones, G., Nedela, R., Škoviera, M.: Chirality groups of maps and hypermaps.J. Algebr. Comb. 29 (2009), 337-355. MR 2496311, 10.1007/s10801-008-0138-z
Reference: [5] d'Azevedo, A. Breda, Nedela, R.: Chiral hypermaps of small genus.Beitr. Algebra Geom. 44 (2003), 127-143. MR 1990988
Reference: [6] Corn, D., Singerman, D.: Regular hypermaps.Eur. J. Comb. 9 (1988), 337-351. Zbl 0665.57002, MR 0950053, 10.1016/S0195-6698(88)80064-7
Reference: [7] Coxeter, H. S. M., Moser, W. O. J.: Generators and Relations for Discrete Groups 4th ed.Ergebnisse der Mathematik und ihrer Grenzgebiete, Bd. 14. Springer-Verlag, Berlin/Heidelberg/New York (1980). Zbl 0422.20001, MR 0562913
Reference: [8] Garbe, D.: Über die regulären Zerlegungen geschlossener orientierbarer Flächen.J. Reine Angew. Math. 237 (1969), 39-55. Zbl 0176.22203, MR 0246198
Reference: [9] Johnson, D. L.: Presentations of Groups.London Mathematical Society Student Texts 15, Cambridge University Press, Cambridge (1990). Zbl 0696.20027, MR 1056695
Reference: [10] Jones, G. A.: Graph imbeddings, groups, and Riemann surfaces.Algebraic methods in graph theory, Conf. Szeged 1978, Colloq. Math. Soc. Janos Bolyai 25, North-Holland, Amsterdam (1981), 297-311. Zbl 0473.05028, MR 0642049
Reference: [11] Jones, G. A., Singerman, D.: Maps, hypermaps and triangle groups.The Grothendieck theory of dessins d'enfants. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 200 (1994), 115-145. Zbl 0833.20045, MR 1305395
Reference: [12] Nostrand, B., Schulte, E., Weiss, A. I.: Constructions of chiral polytopes.Congr. Numerantium 97 (1993), 165-170. Zbl 0812.51017, MR 1267328
Reference: [13] Schulte, E., Weiss, A. I.: Chiral polytopes.Applied geometry and discrete mathematics, Festschr. Victor Klee, DIMACS Ser. Discret. Math. Theor. Comput. Sci. 4 (1991), 493-516. Zbl 0741.51019, MR 1116373
Reference: [14] Sherk, F. A.: A family of regular maps of type $\{6,6\}$.Can. Math. Bull. 5 (1962), 13-20. Zbl 0101.40902, MR 0137030, 10.4153/CMB-1962-003-7
Reference: [15] Thomson, W.: The Robert Boyle Lecture, Oxford University Junior Scientific Club.May 16, 1893, reprinted in Baltimore Lectures (C. J. Clay & Sons, London, 1904).
Reference: [16] Walsh, T. R. S.: Hypermaps versus bipartite maps.J. Comb. Theory Ser. B 18 (1975), 155-163. Zbl 0302.05101, MR 0360328, 10.1016/0095-8956(75)90042-8
Reference: [17] Wilson, S. E.: The smallest nontoroidal chiral maps.J. Graph Theory 2 (1978), 315-318. Zbl 0407.05027, MR 0512802, 10.1002/jgt.3190020405
.

Files

Files Size Format View
CzechMathJ_61-2011-4_14.pdf 279.0Kb application/pdf View/Open
Back to standard record
Partner of
EuDML logo