Hostname: page-component-78c5997874-ndw9j Total loading time: 0 Render date: 2024-11-18T10:40:17.722Z Has data issue: false hasContentIssue false
  • English
  • Français

NEW CONSTRUCTIONS OF SELF-COMPLEMENTARY CAYLEY GRAPHS

Part of: Permutation groups Graph theory

Published online by Cambridge University Press:  12 January 2021

CAI HENG LI ,
GUANG RAO  and
SHU JIAO SONG Open the ORCID record for SHU JIAO SONG [Opens in a new window]
CAI HENG LI
Affiliation:
Southern University of Science and Technology, Shenzhen, China e-mail: lich@sustc.edu.cn
GUANG RAO
Affiliation:
The Chinese University of Hong Kong, Shenzhen, China e-mail: raoguang@cuhk.edu.cn
SHU JIAO SONG
Affiliation:
Yantai University, Yantai, China e-mail: shujiao.song@ytu.edu.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Vertex-primitive self-complementary graphs were proved to be affine or in product action by Guralnick et al. [‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc.356 (2004), 4857–4872]. The product action type is known in some sense. In this paper, we provide a generic construction for the affine case and several families of new self-complementary Cayley graphs are constructed.

Keywords

Cayley graphmetacirculant graphself-complementary graph

MSC classification

Primary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.)
Secondary: 20B05: General theory for finite groups
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 111 , Issue 3 , December 2021 , pp. 372 - 385
Check for updates
Copyright
© 2021 Australian Mathematical Publishing Association Inc.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

Communicated by Michael Giudici

This work was partially supported by NSFC (Nos. 61771019, 11701497 and 11771200) and NSFS (No. ZR2017MA022).

References

Alspach, B., Morris, J. and Vilfred, V., ‘Self-complementary circulant graphs’, Ars Combin. 53 (1999), 187–191.Google Scholar
Beezer, R. A., ‘Sylow subgraphs in self-complementary vertex transitive graphs’, Expo. Math. 24(2) (2006), 185–194.10.1016/j.exmath.2005.09.003CrossRefGoogle Scholar
Berkovic, J. G., ‘Finite metacyclic groups’, Sakharth. SSR Mecn. Akad. Moambe 68 (1972), 529–532.Google Scholar
Biggs, N., Algebraic Graph Theory, 2nd edn, Cambridge Mathematical Library (Cambridge University Press, Cambridge, 1993). ISBN: 0-521-45897-8.Google Scholar
Chvátal, V., Erdös, P. and Hedrlín, Z., ‘Ramsey’s theorem and self-complementary graphs’, Discrete Math. 3 (1972), 301–304.10.1016/0012-365X(72)90087-8CrossRefGoogle Scholar
Clapham, C. R. J., ‘A class of self-complementary graphs and lower bounds of some Ramsey numbers’, J. Graph Theory 3 (1979), 287–289.10.1002/jgt.3190030311CrossRefGoogle Scholar
Figueroa, R. and Giudici, R. E., ‘Group generation of self-complementary graphs’, in: Combinatorics and Graph Theory (Hefei, 1992) (World Scientific, River Edge, NJ, 1993), 131–140.Google Scholar
Fronček, D., Rosa, A. and Širáň, J., ‘The existence of selfcomplementary circulant graphs’, European J. Combin. 17 (1996), 625–628.10.1006/eujc.1996.0053CrossRefGoogle Scholar
Gorenstein, D. and Herstein, I. N., ‘Finite groups admitting a fixed-point-free automorphism of order 4’, Amer. J. Math. 83 (1961), 71–78.10.2307/2372721CrossRefGoogle Scholar
Gulden, F. and Tomasta, P., ‘New lower bounds of some diagonal Ramsey numbers’, J. Graph Theory 7 (1983), 149–151.10.1002/jgt.3190070118CrossRefGoogle Scholar
Guralnick, R., Li, C. H., Praeger, C. E. and Saxl, J., ‘On orbital partitions and exceptionality of primitive permutation groups’, Trans. Amer. Math. Soc. 356 (2004), 4857–4872.10.1090/S0002-9947-04-03396-3CrossRefGoogle Scholar
Hempel, C. E., ‘Metacyclic groups’, Comm. Algebra 28(8) (2000), 3865–3897.10.1080/00927870008827063CrossRefGoogle Scholar
Huhro, E. I., ‘Finite groups that admit a 2-automorphism without fixed point’, Mat. Zametki 23(5) (1978), 651–657.Google Scholar
Jajcay, R. and Li, C. H., ‘Constructions of self-complementary circulants with no multiplicative isomorphisms’, European J. Combin. 22 (2001), 1093–1100.10.1006/eujc.2001.0529CrossRefGoogle Scholar
Li, C. H. and Praeger, C. E., ‘Self-complementary vertex-transitive graphs need not be Cayley graphs’, Bull. Lond. Math. Soc. 33(6) (2001), 653–661.10.1112/S0024609301008505CrossRefGoogle Scholar
Li, C. H. and Praeger, C. E., ‘On partitioning the orbitals of a transitive permutation group’, Trans. Amer. Math. Soc. 355 (2003), 637–653.10.1090/S0002-9947-02-03110-0CrossRefGoogle Scholar
Li, C. H. and Praeger, C. E., ‘Finite permutation groups with a transitive cyclic subgroup’, J. Algebra 349 (2012), 117–127.10.1016/j.jalgebra.2011.10.021CrossRefGoogle Scholar
Li, C. H. and Rao, G., ‘Self-complementary vertex-transitive graphs of order a product of two primes’, Bull. Aust. Math. Soc. 89(2) (2014), 322–330.10.1017/S0004972713000427CrossRefGoogle Scholar
Li, C. H., Rao, G. and Song, S. J., ‘On finite self-complementary metacirculants’, J. Algebraic Combin. 40 (2014), 1135–1144.10.1007/s10801-014-0522-9CrossRefGoogle Scholar
Li, C. H., Sun, S. H. and Xu, J., ‘Self-complementary circulants of prime-power order’, SIAM J. Discrete Math. 28 (2014), 8–17. 10.1137/120870025CrossRefGoogle Scholar
Liskovets, V. and Pschel, R., ‘Non-Cayley-isomorphic self-complementary circulant graphs’, J. Graph Theory 34 (2000) 128–141.10.1002/1097-0118(200006)34:2<128::AID-JGT2>3.0.CO;2-I3.0.CO;2-I>CrossRefGoogle Scholar
Mathon, R., ‘On self-complementary strongly regular graphs’, Discrete Math. 69 (1988), 263–281.10.1016/0012-365X(88)90055-6CrossRefGoogle Scholar
Muzychuck, M., ‘On Sylow subgraphs of vertex-transitive self-complementary graphs’, Bull. Lond. Math. Soc. 31(5) (1999), 531–533.10.1112/S0024609399005925CrossRefGoogle Scholar
Rao, S. B., ‘On regular and strongly regular selfcomplementary graphs’, Discrete Math. 54 (1985), 73–82.10.1016/0012-365X(85)90063-9CrossRefGoogle Scholar
Rödl, V. and Šiňajová, E., ‘Note on Ramsey numbers and self-complementary graphs’, Math. Slovaca 45 (1995), 155–159.Google Scholar
Sachs, H., ‘Über selbstcomplementäre graphen’, Publ. Math. Debrecen 9 (1962), 270–288.Google Scholar
Suprunenko, D. A., ‘Selfcomplementary graphs’, Cybernetics 21 (1985), 559–567.10.1007/BF01074707CrossRefGoogle Scholar
Zelinka, B., ‘Self-complementary vertex-transitive undirected graphs’, Math. Slovaca 29 (1979), 91–95.Google Scholar
Zhang, H., ‘Self-complementary symmetric graphs’, J. Graph Theory 16 (1992), 1–5.10.1002/jgt.3190160102CrossRefGoogle Scholar