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ON THE ORDER OF ARC-STABILIZERS IN ARC-TRANSITIVE GRAPHS

Part of: Graph theory

Published online by Cambridge University Press:  02 October 2009

GABRIEL VERRET
GABRIEL VERRET*
Affiliation:
Institute of Mathematics, Physics and Mechanics, Jadranska 19, 1000 Ljubljana, Slovenia (email: gabriel.verret@fmf.uni-lj.si)
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Abstract

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Let p be a prime. We say that a transitive action of a group L on a set Ω is p-sub-regular if there exist x,y∈Ω such that 〈Lx,Ly〉=L and LYx≅ℤp, where Y =yLx is the orbit of y under Lx. Our main result is that if Γ is a G-arc-transitive graph and the permutation group induced by the action of Gv on Γ(v) is p-sub-regular, then the order of a G-arc-stabilizer is equal to ps−1 where s≤7, s≠6, and moreover, if p=2, then s≤5. This generalizes a classical result of Tutte on cubic arc-transitive graphs as well as some more recent results. We also give a characterization of p-sub-regular actions in terms of arc-regular actions on digraphs and discuss some interesting examples of small degree.

Keywords

arc-transitive grapharc-stabilizer

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 80 , Issue 3 , December 2009 , pp. 498 - 505
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

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