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Finite graphs of valency 4 and girth 4 admitting half-transitive group actions

Part of: Permutation groups Graph theory

Published online by Cambridge University Press:  09 April 2009

Dragan Marušič  and
Roman Nedela
Dragan Marušič
Affiliation:
IMFM, Oddelek za matematiko, Univerza v Ljubljani, Jadranska 19, 1000 Ljubljana, Slovenija e-mail: [email protected]
Roman Nedela
Affiliation:
Katedra Matematiky, Univerzita Mateja Bela, 975 49 Banská Bystrica, Slovensko e-mail: [email protected]
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Abstract

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Finite graphs of valency 4 and girth 4 admitting ½-transitive group actions, that is, vertex- and edge- but not arc-transitive group actions, are investigated. A graph is said to be ½-transitive if its automorphism group acts ½-transitively. There is a natural orientation of the edge set of a ½-transitive graph induced and preserved by its automorphism group. It is proved that in a finite ½-transitive graph of valency 4 and girth 4 the set of 4-cycles decomposes the edge set in such a way that either every 4-cycle is alternating or every 4-cycle is directed relative to this orientation. In the latter case vertex stabilizers are isomorphic to Z2.

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 73 , Issue 2 , October 2002 , pp. 155 - 170
Copyright
Copyright © Australian Mathematical Society 2002

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