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Li et al. [‘On finite self-complementary metacirculants’, J. Algebraic Combin.40 (2014), 1135–1144] proved that the automorphism group of a self-complementary metacirculant is either soluble or has $\text{A}_{5}$ as the only insoluble composition factor, and gave a construction of such graphs with insoluble automorphism groups (which are the first examples of self-complementary graphs with this property). In this paper, we will prove that each simple group is a subgroup (so is a section) of the automorphism groups of infinitely many self-complementary vertex-transitive graphs. The proof involves a construction of such graphs. We will also determine all simple sections of the automorphism groups of self-complementary vertex-transitive graphs of $4$ -power-free order.

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[1]Alspach, B., Morris, J. and Vilfred, V., ‘Self-complementay circulant graphs’, Ars Combin. 53 (1999), 187191.
[2]Beezer, R. A., ‘Sylow subgraphs in self-complementary vertex transitive graphs’, Expo. Math. 24(2) (2006), 185194.
[3]Biggs, N., Algebraic Graph Theory, 2nd edn (Cambridge University Press, New York, 1992).
[4]Chvátal, V., Erdös, P. and Hedrlín, Z., ‘Ramsey’s theorem and self-complementary graphs’, Discrete Math. 3 (1972), 301304.
[5]Clapham, C. R. J., ‘A class of self-complementary graphs and lower bounds of some Ramsey numbers’, J. Graph Theory 3(3) (1979), 287289.
[6]Dobson, E., ‘On self-complementary vertex-transitive graphs of order a product of distinct primes’, Ars Combin. 71 (2004), 249256.
[7]Figueroa, D. and Giudici, R. E., ‘Group generation of self-complementary graphs’, in: Combinatorics and Graph Theory (Hefei, 1992) (World Scientific, River Edge, NJ, 1993), 131140.
[8]Flannery, D. L. and O’Brien, E. A., ‘Linear groups of small degree over finite fields’, Internat. J. Algebra Comput. 15 (2005), 467502.
[9]Godsil, C. D., ‘On the full automorphism group of a graph’, Combinatorica 1 (1981), 243256.
[10]Guldan, F. and Tomasta, P., ‘New lower bounds of some diagonal Ramsey numbers’, J. Graph Theory 7(1) (1983), 149151.
[11]Guralnick, R. M., Li, C. H., Praeger, C. E. and Saxl, J., ‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc. 356(12) (2004), 48574872.
[12]Li, C. H. and Praeger, C. E., ‘On partitioning the orbitals of a transitive permutation group’, Trans. Amer. Math. Soc. 355(2) (2003), 637653.
[13]Li, C. H. and Qiao, S. H., ‘Finite groups of fourth-free order’, J. Group Theory 16(2) (2013), 275298.
[14]Li, C. H. and Rao, G., ‘Self-complementary vertex-transitive graphs of order a product of two primes’, Bull. Aust. Math. Soc. 89 (2014), 322330.
[15]Li, C. H., Rao, G. and Song, S. J., ‘On finite self-complementary metacirculants’, J. Algebraic Combin. 40 (2014), 11351144.
[16]Mathon, R., ‘On self-complementary strongly regular graphs’, Discrete Math. 69 (1988), 263281.
[17]Muzychuk, M., ‘On Sylow subgraphs of vertex-transitive self-complementary graphs’, Bull. Lond. Math. Soc. 31(5) (1999), 531533.
[18]Robinson, D. J. S., A Course in the Theory of Groups (Springer, New York, 1982).
[19]Rödl, V. and Šiňajová, E., ‘Note on Ramsey numbers and self-complementary graphs’, Math. Slovaca 45(3) (1995), 243249.
[20]Suprunenko, D. A., ‘Selfcomplementary graphs’, Cybernetics 21 (1985), 559567.
[21]Zelinka, B., ‘Self-complementary vertex-transitive undirected graphs’, Math. Slovaca 29 (1979), 9195.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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