Article
MSC:
05C10,
05C25,
20B25,
20F65,
51E30,
57M07,
57M60 | MR 2886255 | Zbl 1249.05086 | DOI: 10.1007/s10587-011-0046-6
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Keywords:
hypermap; regular covering; chirality group; chirality index; toroidal hypermaps
hypermap; regular covering; chirality group; chirality index; toroidal hypermaps
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 $\{6,3\}_{b,c}$.
References:
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