Regular Balanced Cayley Maps for Cyclic, Dihedral and Generalized Quaternion Groups

Wang, Yan; Feng, Rong Quan

doi:10.1007/s10114-004-0455-7

Cited by 28 publications

(23 citation statements)

References 5 publications

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“…The following example classifies regular balanced Cayley maps on the dihedral group D 4p for each prime p. The reflexibility and genus of these maps are also determined. Note that regular balanced Cayley maps on dihedral groups have already been classified in [42].…”

Section: ) Every Regular Balanced Cayley Map On Z 4p Is Isomorphic Tomentioning

confidence: 99%

Regular maps of graphs of order 4p

Jia

Feng

2011

Acta. Math. Sin.-English Ser.

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A 2-cell embedding f : X → S of a graph X into a closed orientable surface S can be described combinatorially by a pair M = (X; ρ) called a map, where ρ is a product of disjoint cycle permutations each of which is the permutation of the arc set of X initiated at the same vertex following the orientation of S. It is well known that the automorphism group of M acts semi-regularly on the arc set of X and if the action is regular, then the map M and the embedding f are called regular. Let p and q be primes. Du et al. [J. Algebraic Combin., 19, 123-141 (2004)] classified the regular maps of graphs of order pq. In this paper all pairwise non-isomorphic regular maps of graphs of order 4p are constructed explicitly and the genera of such regular maps are computed. As a result, there are twelve sporadic and six infinite families of regular maps of graphs of order 4p; two of the infinite families are regular maps with the complete bipartite graphs K2p,2p as underlying graphs and the other four infinite families are regular balanced Cayley maps on the groups Z 4p , Z 2 2 × Z p and D 4p .

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Section: ) Every Regular Balanced Cayley Map On Z 4p Is Isomorphic Tomentioning

confidence: 99%

Regular maps of graphs of order 4p

Jia

Feng

2011

Acta. Math. Sin.-English Ser.

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“…is known to be a regular balanced Cayley map [14]. For our convenience, let T be the set of all triples (n, , k) of positive integers such that < n and k is the smallest integer satisfying…”

Section: Preliminariesmentioning

confidence: 99%

A Classification of Prime-Valent Regular Cayley Maps on Abelian, Dihedral and Dicyclic Groups

Kim¹,

Kwon²,

Lee³

2010

Bulletin of the Korean Mathematical Society

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Abstract. A Cayley map is a 2-cell embedding of a Cayley graph into an orientable surface with the same local orientation induced by a cyclic permutation of generators at each vertex. In this paper, we provide classifications of prime-valent regular Cayley maps on abelian groups, dihedral groups and dicyclic groups. Consequently, we show that all prime-valent regular Cayley maps on dihedral groups are balanced and all prime-valent regular Cayley maps on abelian groups are either balanced or anti-balanced. Furthermore, we prove that there is no prime-valent regular Cayley map on any dicyclic group.

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“…In particular, if m = 1, then M p,2 (m, r) is a dihedral group of order 2p. One may refer to [12] for the classification of the regular balanced Cayley maps of dihedral groups. For the sake of completeness, We restate the result in the following theorem.…”

Section: Lemma 23 ([5]mentioning

confidence: 99%

“…For regular balanced Cayley maps, it has been shown that all odd order abelian groups possess at least one regular balanced Cayley map [4]. Wang and Feng [12] classified all regular balanced Cayley maps for cyclic, dihedral and generalized quaternion groups. In [9], Oh proved the non-existence of regular balanced Cayley maps with semi-dihedral groups.…”

Section: Introductionmentioning

confidence: 99%

“…In this paper, we pay our attentions to the regular balanced Cayley maps of minimal nonabelian metacyclic groups. Since the regular balanced Cayley maps of Q 8 have been classified in [12] (Q 8 has exactly one regular balanced Cayley map up to isomorphism), we only consider the groups M p,q (m, r) and M p (n, m). In Section 3, we show that M p,q (m, r) has regular balanced Cayley maps if and only if q is 2 and we list all of them up to isomorphism.…”

Section: Introductionmentioning

confidence: 99%

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Classification of regular balanced Cayley maps of minimal non-abelian metacyclic groups

Yuan

Wang

2017

Ars Math. Contemp.

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In this paper, we classify the regular balanced Cayley maps of minimal non-abelian metacyclic groups. Besides the quaternion group Q 8 , there are two infinite families of such groups which are denoted by M p,q (m, r) and M p (n, m), respectively. Firstly, we prove that there are regular balanced Cayley maps of M p,q (m, r) if and only if q = 2 and we list all of them up to isomorphism. Secondly, we prove that there are regular balanced Cayley maps of M p (n, m) if and only if p = 2 and n = m or n = m + 1 and there is exactly one such map up to isomorphism in either case. Finally, as a corollary, we prove that any metacyclic p-group for odd prime number p does not have regular balanced Cayley maps.

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Regular Balanced Cayley Maps for Cyclic, Dihedral and Generalized Quaternion Groups

Cited by 28 publications

References 5 publications

Regular maps of graphs of order 4p

Regular maps of graphs of order 4p

A Classification of Prime-Valent Regular Cayley Maps on Abelian, Dihedral and Dicyclic Groups

Classification of regular balanced Cayley maps of minimal non-abelian metacyclic groups

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