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

Published online by Cambridge University Press:  09 April 2009

Dragan Marušič
Affiliation:
IMFM, Oddelek za matematiko, Univerza v Ljubljani, Jadranska 19, 1000 Ljubljana, Slovenija e-mail: dragan.marusic@uni-lj.si
Roman Nedela
Affiliation:
Katedra Matematiky, Univerzita Mateja Bela, 975 49 Banská Bystrica, Slovensko e-mail: nedela@bb.sanet.sk
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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2002

References

[1]Alspach, B., Marušič, D. and Nowitz, L., ‘Constructing graphs which are ½-transitive’, J. Austral. Math. Soc. Ser A 56 (1994), 391402.Google Scholar
[2]Alspach, B. and Xu, M.-Y., ‘½-arc-transitive graphs of order 3p’, J. Algebraic Combin. 3 (1994), 347355.Google Scholar
[3]Biggs, N. and White, A. T., Permutation groups and combinatorial structures (Cambridge University Press, Cambridge, 1979).Google Scholar
[4]Bondy, A. and Murty, U. S. R., Graph theory with applications (American Elsevier, New York, 1976).CrossRefGoogle Scholar
[5]Bouwer, I. Z., ‘Vertex and edge-transitive but not 1-transitive graphs’, Canad. Math. Bull. 13 (1970), 231237.Google Scholar
[6]Coxeter, H. S. M. and Moser, W. O. J., Generators and relations for discrete groups (Springer, New York, 1972).CrossRefGoogle Scholar
[7]Du, S. F. and Xu, M. Y., ‘Vertex-primitive 1/2-arc-transitive graphs of smallest order’, Comm. Algebra 27 (1999), 163171.CrossRefGoogle Scholar
[8]Holt, D. F., ‘A graph which is edge transitive but not arc transitive’, J. Graph Theory 5 (1981), 201204.Google Scholar
[9]Malnič, A. and Marušič, D., ‘Constructing ½-transitive graphs of valency 4 and vertex stabilizer Z2 × Z2’, Discrete Math. 245 (2002), 203216.Google Scholar
[10]Marušič, D., ‘Half-transitive group actions on finite graphs of valency 4’, J. Combin. Theory Ser. B 73 (1998), 4176.CrossRefGoogle Scholar
[11]Marušič, D. and Nedela, R., ‘Maps, one-regular graphs and ½-transitive graphs of valency 4’, European J. Combin. 19 (1998), 345354.CrossRefGoogle Scholar
[12]Marušič, D. and Praeger, C. E., ‘Tetravalent graphs admitting half-transitive actions; alternating cycles’, J. Combin. Theory Ser. B 75 (1999), 188205.Google Scholar
[13]Marušič, D. and Xu, M.-Y., ‘A ½-transitive graph of valency 4 with a nonsolvable group of automorphisms’, J. Graph Theory 25 (1997), 133138.Google Scholar
[14]Šajna, M., ‘Half-transitivity of some metacirculants’, Discrete Math. 185 (1998), 117136.CrossRefGoogle Scholar
[15]Taylor, D. E. atid Xu, M.-Y., ‘Vertex-primitive ½-transitive graphs’, J. Austral. Math. Soc. Ser. A57 (1994), 113124.CrossRefGoogle Scholar
[16]Tutte, W. T., Connectivity in graphs (University of Toronto Press, Toronto, 1966).CrossRefGoogle Scholar
[17]Wang, R. J., ‘Half-transitive graphs of order a product of two distinct primes’, Comm. Algebra 22 (1994), 917927.Google Scholar
[18]Wielandt, H., Finite permutation groups (Academic Press, New York, 1964).Google Scholar
[19]Xu, M.-Y., ‘Half-transitive graphs of prime cube order’, J. Algebraic Combin. 1 (1992), 275282.Google Scholar