Journal article

On subgroup perfect codes in Cayley graphs

Junyang Zhang, Sanming Zhou

EUROPEAN JOURNAL OF COMBINATORICS | ACADEMIC PRESS LTD- ELSEVIER SCIENCE LTD | Published : 2021

Abstract

A perfect code in a graph Γ=(V,E) is a subset C of V such that no two vertices in C are adjacent and every vertex in V∖C is adjacent to exactly one vertex in C. A subgroup H of a group G is called a subgroup perfect code of G if there exists a Cayley graph of G which admits H as a perfect code. Equivalently, H is a subgroup perfect code of G if there exists an inverse-closed subset A of G containing the identity element such that (A,H) is a tiling of G in the sense that every element of G can be uniquely expressed as the product of an element of A and an element of H. In this paper we obtain multiple results on subgroup perfect codes of finite groups, including a few necessary and sufficient..

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University of Melbourne Researchers

Grants

Awarded by National Natural Science Foundation of China


Awarded by Basic Research and Frontier Exploration Project of Chongqing, China


Awarded by Science and Technology Research Program of Chongqing Municipal Education Commission, China


Funding Acknowledgements

We are grateful to the two anonymous referees whose comments and suggestions led to significant improvements of this paper. The first author thanks The University of Melbourne for its hospitality where part of this work was done during his one-year visit. The first author was supported by the National Natural Science Foundation of China (No. 11671276), the Basic Research and Frontier Exploration Project of Chongqing, China (No. cstc2018jcyjAX0010), and the Science and Technology Research Program of Chongqing Municipal Education Commission, China (No. KJQN201800512). The second author was supported by the National Natural Science Foundation of China (No. 61771019) and the Research Grant Support Scheme of The University of Melbourne, Australia.