{"title":"A sharp lower bound on the independence number of k-regular connected hypergraphs with rank R","authors":"Zhongzheng Tang, Haoyang Zou, Zhuo Diao","doi":"10.1007/s00236-024-00468-6","DOIUrl":null,"url":null,"abstract":"<div><p>Let <i>H</i>(<i>V</i>, <i>E</i>) be a <i>k</i>-regular connected hypergraph with rank <i>R</i> on <i>n</i> vertices and <i>m</i> edges. A set of vertices <span>\\(S\\subseteq V\\)</span> is an independent set if every two vertices in <i>S</i> are not adjacent. The independence number is the maximum cardinality of an independent set, denoted by <span>\\(\\alpha (H)\\)</span>. In this paper, we prove the following inequality: <span>\\(\\alpha (H)\\ge \\frac{m-(k-2)n-1}{R}\\)</span>, and the equality holds if and only if <i>H</i> is a hypertree with <i>R</i>-perfect matching. Based on the proofs, some combinatorial algorithms on the independence number are designed.</p></div>","PeriodicalId":7189,"journal":{"name":"Acta Informatica","volume":"62 1","pages":""},"PeriodicalIF":0.4000,"publicationDate":"2024-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Acta Informatica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00236-024-00468-6","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, INFORMATION SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
Let H(V, E) be a k-regular connected hypergraph with rank R on n vertices and m edges. A set of vertices \(S\subseteq V\) is an independent set if every two vertices in S are not adjacent. The independence number is the maximum cardinality of an independent set, denoted by \(\alpha (H)\). In this paper, we prove the following inequality: \(\alpha (H)\ge \frac{m-(k-2)n-1}{R}\), and the equality holds if and only if H is a hypertree with R-perfect matching. Based on the proofs, some combinatorial algorithms on the independence number are designed.
期刊介绍：
Acta Informatica provides international dissemination of articles on formal methods for the design and analysis of programs, computing systems and information structures, as well as related fields of Theoretical Computer Science such as Automata Theory, Logic in Computer Science, and Algorithmics.
Topics of interest include:
• semantics of programming languages
• models and modeling languages for concurrent, distributed, reactive and mobile systems
• models and modeling languages for timed, hybrid and probabilistic systems
• specification, program analysis and verification
• model checking and theorem proving
• modal, temporal, first- and higher-order logics, and their variants
• constraint logic, SAT/SMT-solving techniques
• theoretical aspects of databases, semi-structured data and finite model theory
• theoretical aspects of artificial intelligence, knowledge representation, description logic
• automata theory, formal languages, term and graph rewriting
• game-based models, synthesis
• type theory, typed calculi
• algebraic, coalgebraic and categorical methods
• formal aspects of performance, dependability and reliability analysis
• foundations of information and network security
• parallel, distributed and randomized algorithms
• design and analysis of algorithms
• foundations of network and communication protocols.
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