Czechoslovak Mathematical Journal, Vol. 61, No. 4, pp. 1037-1047, 2011

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On chirality groups and regular coverings

of regular oriented hypermaps

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Antonio Breda d'Azevedo, Ilda Inácio Rodrigues, Maria Elisa Fernandes

* Antonio Breda D'Azevedo*, University of Aveiro, Aveiro, Portugal,
e-mail: ` breda@ua.pt`;
*Ilda Inacio Rodrigues*, University of Beira Interior,
Covilhã, Portugal, e-mail: ` ilda@ubi.pt`; * Maria Elisa Fernandes*, University of Aveiro, Aveiro, Portugal, e-mail: ` maria.elisa@ua.ptz`

**Abstract:** 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 $\{6,3\}_{b,c}$.

**Keywords:** hypermap, regular covering, chirality group, chirality index, toroidal hypermaps

**Classification (MSC 2010):** 05C10, 05C25, 20B25, 20F65, 51E30, 57M07, 57M60

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