Abstract
Let \(\Gamma \) be a graph with vertex set V. If a subset C of V is independent in \(\Gamma \) and every vertex in \(V\setminus C\) is adjacent to exactly one vertex in C, then C is called a perfect code of \(\Gamma \). Let G be a finite group and let S be a square-free normal subset of G. The Cayley sum graph of G with respect to S is a simple graph with vertex set G and two vertices x and y are adjacent if \(xy\in S\). A subset C of G is called a perfect code of G if there exists a Cayley sum graph of G which admits C as a perfect code. In particular, if a subgroup of G is a perfect code of G, then the subgroup is called a subgroup perfect code of G. In this paper, we give a necessary and sufficient condition for a non-trivial subgroup of an abelian group with non-trivial Sylow 2-subgroup to be a subgroup perfect code of the group. This reduces the problem of determining when a given subgroup of an abelian group is a perfect code to the case of abelian 2-groups. As an application, we classify the abelian groups whose every non-trivial subgroup is a subgroup perfect code. Moreover, we determine all subgroup perfect codes of a cyclic group, a dihedral group and a generalized quaternion group.
Similar content being viewed by others
References
Alon N.: Large sets in finite fields are sumsets. J. Number Theory 126, 110–118 (2007).
Amooshahi M., Taeri B.: On Cayley sum graphs of non-abelian groups. Graphs Combin. 32, 17–29 (2016).
Biggs N.L.: Perfect codes in graphs. J. Combin. Theory Ser. B 15, 289–296 (1973).
Cheyne B., Gupta V., Wheeler C.: Hamilton cycles in addition graphs. Rose-Hulman Undergrad. Math J. 4, 1–17 (2003). (electronic).
Chung F.R.K.: Diameters and eigenvalues. J. Am. Math. Soc. 2, 187–196 (1989).
Dejter I.J., Serra O.: Efficient dominating sets in Cayley graphs. Discrete Appl. Math. 129, 319–328 (2003).
Delsarte P.: An algebraic approach to the association schemes of coding theory. Philips Res. Rep. Suppl. 10, 165, 171, 173 (1973).
Deng Y.-P., Sun Y.-Q., Liu Q., Wang H.-C.: Efficient dominating sets in circulant graphs. Discret. Math. 340, 1503–1507 (2017).
DeVos M., Goddyn L., Mohar B., S̆ámal R.: Cayley sum graphs and eigenvalues of \((3,6)\)-fullerenes. J. Combin. Theory Ser. B 99, 358–369 (2009).
Feng R., Huang H., Zhou S.: Perfect codes in circulant graphs. Discret. Math. 340, 1522–1527 (2017).
Grynkiewicz D., Levb V.F., Serra O.: Connectivity of addition Cayley graphs. J. Combin. Theory Ser. B 99, 202–217 (2009).
Heden O.: A survey of perfect codes. Adv. Math. Commun. 2, 223–247 (2008).
Huang H., Xia B., Zhou S.: Perfect codes in Cayley graphs. SIAM J. Discret. Math. 32, 548–559 (2018).
James G., Liebeck M.: Representations and Characters of Groups. Cambridge University Press, Cambridge (2001).
Johnson D.L.: Topics in the Theory of Group Presentations. London Mathematical Society Lecture Note Series, vol. 42. Cambridge University Press, Cambridge (1980).
Konyagin S.V., Shkredov I.D.: On subgraphs of random Cayley sum graphs. Eur. J. Comb. 70, 61–74 (2018).
Kratochvíl J.: Perfect codes over graphs. J. Comb. Theory Ser. B 40, 224–228 (1986).
Lee J.: Independent perfect domination sets in Cayley graphs. J. Graph Theory 37, 213–219 (2001).
Lev V.F.: Sums and differences along Hamiltonian cycles. Discret. Math. 310, 575–584 (2010).
Li C.-K., Nelson I.: Perfect codes on the towers of Hanoi graph. Bull. Aust. Math. Soc. 57, 367–376 (1998).
Ma X., Wang K.: Integral Cayley sum graphs and groups. Discuss. Math. Graph Theory 36, 797–803 (2016).
Mollard M.: On perfect codes in Cartesian products of graphs. Eur. J. Comb. 32, 398–403 (2011).
van Lint J.H.: A survey of perfect codes. Rocky Mt. J. Math. 5, 199–224 (1975).
Žerovnik J.: Perfect codes in direct products of cycles—a complete characterization. Adv. Appl. Math. 41, 197–205 (2008).
Zhou S.: Total perfect codes in Cayley graphs. Des. Codes Cryptogr. 81, 489–504 (2016).
Zhou S.: Cyclotomic graphs and perfect codes. J. Pure Appl. Algebra 223, 931–947 (2019).
Acknowledgements
We are grateful to the anonymous referees for their helpful comments which led to improvement of Section 3 and betterment of presentation of the paper. Ma was supported by the National Natural Science Foundation of China (Grant Nos. 11801441 and 61976244), the Natural Science Basic Research Program of Shaanxi (Program No. 2020JQ-761), and the Young Talent fund of University Association for Science and Technology in Shaanxi, China (Grant No. 20190507). Feng was supported by the National Natural Science Foundation of China (11701281), the Natural Science Foundation of Jiangsu Province (BK20170817) and the Grant of China Postdoctoral Science Foundation. Wang was supported by the National Natural Science Foundation of China (11671043).
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. A. Zinoviev.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Ma, X., Feng, M. & Wang, K. Subgroup perfect codes in Cayley sum graphs. Des. Codes Cryptogr. 88, 1447–1461 (2020). https://doi.org/10.1007/s10623-020-00758-3
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10623-020-00758-3