225 research outputs found
Finite edge-transitive dihedrant graphs
AbstractIn this paper, we first prove that each biquasiprimitive permutation group containing a regular dihedral subgroup is biprimitive, and then give a classification of such groups. The classification is then used to classify vertex-quasiprimitive and vertex-biquasiprimitive edge-transitive dihedrants. Moreover, a characterization of valencies of normal edge-transitive dihedrants is obtained, and some classes of examples with certain valences are constructed
Using mixed dihedral groups to construct normal Cayley graphs, and a new bipartite -arc-transitive graph which is not a Cayley graph
A \emph{mixed dihedral group} is a group with two disjoint subgroups
and , each elementary abelian of order , such that is generated by
, and . In this paper we give a sufficient
condition such that the automorphism group of the Cayley graph \Cay(H,(X\cup
Y)\setminus\{1\}) is equal to , where is the setwise
stabiliser in \Aut(H) of . We use this criterion to resolve a
questions of Li, Ma and Pan from 2009, by constructing a -arc transitive
normal cover of order of the complete bipartite graph \K_{16,16} and
prove that it is \emph{not} a Cayley graph.Comment: arXiv admin note: text overlap with arXiv:2303.00305,
arXiv:2211.1680
The vertex-transitive TLF-planar graphs
We consider the class of the topologically locally finite (in short TLF)
planar vertex-transitive graphs, a class containing in particular all the
one-ended planar Cayley graphs and the normal transitive tilings. We
characterize these graphs with a finite local representation and a special kind
of finite state automaton named labeling scheme. As a result, we are able to
enumerate and describe all TLF-planar vertex-transitive graphs of any given
degree. Also, we are able decide to whether any TLF-planar transitive graph is
Cayley or not.Comment: Article : 23 pages, 15 figures Appendix : 13 pages, 72 figures
Submitted to Discrete Mathematics The appendix is accessible at
http://www.labri.fr/~renault/research/research.htm
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