Abstract
A factorisation of a complete graph K n is a partition of its edges with each part corresponding to a spanning subgraph (not necessarily connected), called a factor. A factorisation is called homogeneous if there are subgroups M<G≤S n such that M is vertex-transitive and fixes each factor setwise, and G permutes the factors transitively. We classify the homogeneous factorisations of K n for which there are such subgroups G,M with M transitive on the edges of a factor as well as the vertices. We give infinitely many new examples.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Alspach, B., Morris, J., Vilfred, V.: Self-complementary circulant graphs. Ars Combinatoria 53, 187–191 (1999)
Biggs, N.: Algebraic Graph Theory. Cambridge University Press, Cambridge (1993)
Bonisoli, A., Buratti, M., Mazzuoccolo, G.: Doubly transitive 2-factorizations. J. Combin. Designs 15(2), 120–132 (2007)
Bonisoli, A., Labbate, D.: One-factorizations of complete graphs with vertex-regular automorphism groups. J. Combin. Designs 10(1), 1–16 (2002)
Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance Regular Graphs. Springer, Berlin (1989)
Cameron, P.J.: Finite permutation groups and finite simple groups. Bull. London Math. Soc. 13, 1–22 (1981)
Cameron, P.J.: Combinatorics: Topics, Techniques, Algorithms. Cambridge University Press, Cambridge (1994)
Cannon, J., Playoust, C.: An Introduction to Magma. School of Mathematics and Statistics, University of Sydney. (See also http://magma.maths.usyd.edu.au/)
Dixon, J.D., Mortimer, B.: Permutation Groups. Springer, New York (1996)
Foulser, D.A.: The flag-transitive collineation group of the finite Desarguesian affine planes. Canad. J. Math. 16, 443–472 (1964)
Foulser, D.A., Kallaher, M.J.: Solvable, flag-transitive, rank 3 collineation groups. Geom. Ded. 7, 111–130 (1978)
Giudici, M., Li, C.H., Potočnik, P., Praeger, C.E.: Homogeneous factorisations of graphs and digraphs. European J. Combin. 27, 11–37 (2006)
Harary, F., Robinson, R.W., Wormald, N.C.: Isomorphic factorisations I: Complete graphs. Trans. Amer. Math. Soc. 242, 243–260 (1978)
Harary, F., Robinson, R.W.: Isomorphic factorisations X: Unsolved problems. J. Graph Theory 9, 67–86 (1985)
Huppert, B.: Endliche Gruppen I. Springer, Berlin (1967)
Huppert, B., Blackburn, N.: Finite Groups III. Springer, Berlin (1982)
Jajcay, R., Li, C.H.: Constructions of self-complementary circulants with no multiplicative isomorphisms. European J. Combin. 22, 1093–1100 (2001)
Kantor, W.M.: Automorphism groups of designs. Math. Z. 109, 246–252 (1969)
Kantor, W.M.: Homogeneous designs and geometric lattices. J. Combin. Theory Ser. A 38, 66–74 (1985)
Li, C.H.: On self-complementary vertex-transitive graphs. Comm. Algebra 25, 3903–3908 (1997)
Li, C.H., Praeger, C.E.: On partitioning the orbitals of a transitive permutation group. Trans. Amer. Math. Soc. 355, 637–653 (2003)
Liebeck, M.W.: The affine permutation groups of rank 3. Proc. London Math. Soc. 54, 477–516 (1987)
Lim, T.K.: Arc-transitive homogeneous factorizations and affine planes. J. Combin. Designs 14, 290–300 (2008)
Lim, T.K.: Edge-transitive homogeneous factorisations of complete graphs. PhD thesis, University of Western Australia (2004). (http://theses.library.uwa.edu.au/adt-WU2004.0039/)
Lim, T.K., Praeger, C.E.: On generalised Paley graphs and their automorphism groups. Mich. Math. J. (2008, to appear)
Muzychuk, M.: On Sylow’s subgraphs of vertex-transitive self-complementary graphs. Bull. London Math. Soc. 31, 531–533 (1999)
Peisert, W.: All self-complementary symmetric graphs. J. Algebra 240, 209–229 (2001)
Praeger, C.E.: Finite transitive permutation groups and finite vertex-transitive graphs. In: Graph Symmetry (Montrael, PQ, 1996). Kluwer Academic, Dordrecht (1997)
Sibley, T.Q.: On classifying finite edge colored graphs with two transitive automorphism groups. J. Combin. Theory Ser. B 90, 121–138 (2004)
Suprunenko, D.A.: Self-complementary graphs. Cybernetics 21, 559–567 (1985)
Suzuki, M.: Group Theory I. Springer, New York (1982)
Tutte, W.T.: Connectivity in Graphs. University of Toronto Press, Toronto (1966)
Wielandt, H.: Finite Permutation Groups. Academic Press, New York (1964)
Zhang, H.: Self-complementary symmetric graphs. J. Graph Theory 16, 1–5 (1992)
Zsigmondy, K.: Zur Theorie der Potenzreste. Monatsh. Math. Phys. 3, 265–284 (1892)
Author information
Authors and Affiliations
Corresponding author
Additional information
This paper forms part of an Australian Research Council Discovery Grant project, and was a major part of the PhD project of the second author.
Rights and permissions
About this article
Cite this article
Li, C.H., Lim, T.K. & Praeger, C.E. Homogeneous factorisations of complete graphs with edge-transitive factors. J Algebr Comb 29, 107–132 (2009). https://doi.org/10.1007/s10801-008-0127-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-008-0127-2