A Classification of Prime-valent Regular Cayley Maps on Some Groups
نویسندگان
چکیده
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.
منابع مشابه
A Classification of Prime-valent Regular Cayley Maps on Abelian, Dihedral and Dicyclic Groups
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 ba...
متن کاملNORMAL 6-VALENT CAYLEY GRAPHS OF ABELIAN GROUPS
Abstract : We call a Cayley graph Γ = Cay (G, S) normal for G, if the right regular representation R(G) of G is normal in the full automorphism group of Aut(Γ). In this paper, a classification of all non-normal Cayley graphs of finite abelian group with valency 6 was presented.
متن کاملOn symmetries of Cayley graphs and the graphs underlying regular maps
By definition, Cayley graphs are vertex-transitive, and graphs underlying regular or orientably-regular maps (on surfaces) are arc-transitive. This paper addresses questions about how large the automorphism groups of such graphs can be. In particular, it is shown how to construct 3-valent Cayley graphs that are 5-arc-transitive (in answer to a question by Cai Heng Li), and Cayley graphs of vale...
متن کاملArc–transitive Non–cayley Graphs from Regular Maps
We prove that the underlying graphs of p-gonal r-valent orientably regular maps are arc-transitive but non-Cayley if r ≥ 3 and p is a prime greater than r(r − 1).
متن کاملArc { Transitive Non { Cayley Graphs from Regular
We prove that the underlying graphs of p-gonal r-valent orientably regular maps are arc-transitive but non-Cayley if r 3 and p is a prime greater than r(r ? 1).
متن کامل