Journal article

SUBGROUP PERFECT CODES IN CAYLEY GRAPHS

Xuanlong Ma, Gary L Walls, Kaishun Wang, Sanming Zhou

SIAM JOURNAL ON DISCRETE MATHEMATICS | SIAM PUBLICATIONS | Published : 2020

Abstract

Let Γ be a graph with vertex set V (Γ). A subset C of V (Γ) is called a perfect code in Γ if C is an independent set of Γ and every vertex in V (Γ) \ C is adjacent to exactly one vertex in C. A subset C of a group G is called a perfect code of G if there exists a Cayley graph of G which admits C as a perfect code. A group G is said to be code-perfect if every proper subgroup of G is a perfect code of G. In this paper we prove that a group is code-perfect if and only if it has no elements of order 4. We also prove that a proper subgroup H of an abelian group G is a perfect code of G if and only if the Sylow 2-subgroup of H is a perfect code of the Sylow 2-subgroup of G. This reduces the probl..

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University of Melbourne Researchers

Grants

Awarded by National Natural Science Foundation of China


Awarded by Natural Science Basic Research Program of Shaanxi


Funding Acknowledgements

The first author was supported by the National Natural Science Foundation of China (grant 11801441) and by the Natural Science Basic Research Program of Shaanxi (program 2020JQ761). The third author was supported by the National Natural Science Foundation of China (grant 11671043). The fourth author was supported by the National Natural Science Foundation of China (grant 61771019) and by the Research Grant Support Scheme of The University of Melbourne.