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ON GRAPHS OF PRIME VALENCY ADMITTING A SOLVABLE ARC-TRANSITIVE GROUP

Published online by Cambridge University Press:  13 May 2015

BOŠTJAN KUZMAN*
Affiliation:
University of Ljubljana, Faculty of Education, Department of Math and Computer Science, Kardeljeva ploščad 16, 1000 Ljubljana, Slovenia Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia email bostjan.kuzman@pef.uni-lj.si
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Abstract

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Let $X$ be a simple, connected, $p$-valent, $G$-arc-transitive graph, where the subgroup $G\leq \text{Aut}(X)$ is solvable and $p\geq 3$ is a prime. We prove that $X$ is a regular cover over one of the three possible types of graphs with semi-edges. This enables short proofs of the facts that $G$ is at most 3-arc-transitive on $X$ and that its edge kernel is trivial. For pentavalent graphs, two further applications are given: all $G$-basic pentavalent graphs admitting a solvable arc-transitive group are constructed and an example of a non-Cayley graph of this kind is presented.

Type
Research Article
Copyright
© 2015 Australian Mathematical Publishing Association Inc. 

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