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One-regular cubic graphs of order a small number times a prime or a prime square

Part of: Graph theory Permutation groups

Published online by Cambridge University Press:  09 April 2009

Yan-Quan Feng  and
Jin Ho Kwak
Yan-Quan Feng
Affiliation:
Department of Mathematics, Beijing Jiaotong University, Beijing 100044, P.R. China, e-mail: [email protected]
Jin Ho Kwak
Affiliation:
Department of Mathematics, Pohang University of Science, and Technology, Pohang, 790–784 Korea e-mail: [email protected]
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Abstract

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A graph is one-regular if its automorphism group acts regularly on the set of its arcs. In this paper we show that there exists a one-regular cubic graph of order 2p or 2p2 where p is a prime if and only if 3 is a divisor of p – 1 and the graph has order greater than 25. All of those one-regular cubic graphs are Cayley graphs on dihedral groups and there is only one such graph for each fixed order. Surprisingly, it can be shown that there is no one-regular cubic graph of order 4p or 4p2.

Keywords

Cayley grapharc-transitive graphone-regular graph

MSC classification

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) 20B25: Finite automorphism groups of algebraic, geometric, or combinatorial structures
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 345 - 356
Copyright
Copyright © Australian Mathematical Society 2004

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