Perfect codes in proper reduced power graphs of finite groups

Abstract

Let Γ be a graph with vertex set $V(\Gamma ).$ A subset C of $V(\Gamma )$ is a perfect code of Γ if C is an independent set such that every vertex in $V(\Gamma )\setminus C$ is adjacent to exactly one vertex in C. A subset T of $V(\Gamma )$ is a total perfect code of Γ if every vertex of Γ is adjacent to exactly one vertex in T. Let G be a finite group. The proper reduced power graph of G is the undirected graph whose vertex set consists of all nonidentity elements, and two distinct vertices x and y are adjacent if $〈x〉\subset 〈y〉$ or $〈y〉\subset 〈x〉.$ In this paper, we give a necessary and sufficient condition for a proper reduced power graph to contain a perfect code. In particular, we determine all perfect codes of a proper reduced power graph provided that the proper reduced power graph admits a perfect code. Moreover, we give some necessary conditions for a proper reduced power graph to contain a total perfect code. As applications, we determine the abelian groups and generalized quaternion groups whose proper reduced power graphs admit a total perfect code. We also characterize all finite groups whose proper reduced power graphs admit a total perfect code of size 2.

Keywords:

• Finite group
• perfect code
• proper reduced power graph
• total perfect code

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 05C25
• 05C69

Acknowledgements

The author is grateful to the referee for useful suggestions and comments.

Notes

1 See http://www.planetmath.org/GeneralizedQuaternionGroup.

Additional information

Funding

This research 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).