Geodesic
On chirality groups and regular coverings of regular oriented hypermaps
Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1037-1047
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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 $\{6,3\}_{b,c}$.
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 $\{6,3\}_{b,c}$.
DOI : 10.1007/s10587-011-0046-6
Classification : 05C10, 05C25, 20B25, 20F65, 51E30, 57M07, 57M60
Keywords: hypermap; regular covering; chirality group; chirality index; toroidal hypermaps 
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     title = {On chirality groups and regular coverings of regular oriented hypermaps},
     journal = {Czechoslovak Mathematical Journal},
     pages = {1037--1047},
     year = {2011},
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Breda d'Azevedo, Antonio; Rodrigues, Ilda Inácio; Fernandes, Maria Elisa. On chirality groups and regular coverings of regular oriented hypermaps. Czechoslovak Mathematical Journal, Tome 61 (2011) no. 4, pp. 1037-1047. doi: 10.1007/s10587-011-0046-6

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