{"title":"Perfect codes of bi-Cayley graphs","authors":"Yan Wang, Kai Yuan, Jian-Xun Li","doi":"10.1016/j.jcta.2025.106079","DOIUrl":null,"url":null,"abstract":"<div><div>A bi-Cayley graph is a graph that has a semiregular group <em>H</em> of automorphisms having exactly two orbits on vertices, and it is called an algebraically Cayley graph if its full automorphism group contains a regular subgroup <em>G</em> such that <em>H</em> is a subgroup of <em>G</em>. An independent vertex subset of a graph is called a perfect code if each vertex outside of this subset is adjacent to exactly one vertex in it. In this paper, we give a necessary and sufficient condition for a bi-Cayley graph to be an algebraically Cayley graph, and perfect codes of such bi-Cayley graphs can be determined by the theory of perfect codes in Cayley graphs. Equivalent conditions for subsets to be perfect codes of regular (in terms of graph theory) bi-Cayley graphs are also given.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"217 ","pages":"Article 106079"},"PeriodicalIF":1.2000,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Theory Series A","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0097316525000743","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
A bi-Cayley graph is a graph that has a semiregular group H of automorphisms having exactly two orbits on vertices, and it is called an algebraically Cayley graph if its full automorphism group contains a regular subgroup G such that H is a subgroup of G. An independent vertex subset of a graph is called a perfect code if each vertex outside of this subset is adjacent to exactly one vertex in it. In this paper, we give a necessary and sufficient condition for a bi-Cayley graph to be an algebraically Cayley graph, and perfect codes of such bi-Cayley graphs can be determined by the theory of perfect codes in Cayley graphs. Equivalent conditions for subsets to be perfect codes of regular (in terms of graph theory) bi-Cayley graphs are also given.
期刊介绍:
The Journal of Combinatorial Theory publishes original mathematical research concerned with theoretical and physical aspects of the study of finite and discrete structures in all branches of science. Series A is concerned primarily with structures, designs, and applications of combinatorics and is a valuable tool for mathematicians and computer scientists.