Balanced Independent Sets and Colorings of Hypergraphs

IF 1 3区数学 Q2 MATHEMATICS Journal of Graph Theory Pub Date : 2025-01-06 DOI:10.1002/jgt.23212

Abhishek Dhawan

{"title":"Balanced Independent Sets and Colorings of Hypergraphs","authors":"Abhishek Dhawan","doi":"10.1002/jgt.23212","DOIUrl":null,"url":null,"abstract":"A <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0001\" wiley:location=\"equation/jgt23212-math-0001.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>-uniform hypergraph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n \n <mo>=</mo>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>V</mi>\n \n <mo>,</mo>\n \n <mi>E</mi>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0002\" wiley:location=\"equation/jgt23212-math-0002.png\"><mrow><mrow><mi>H</mi><mo>=</mo><mrow><mo>(</mo><mrow><mi>V</mi><mo>,</mo><mi>E</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> is <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0003\" wiley:location=\"equation/jgt23212-math-0003.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>-partite if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>V</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0004\" wiley:location=\"equation/jgt23212-math-0004.png\"><mrow><mrow><mi>V</mi></mrow></mrow></math></annotation>\n </semantics></math> can be partitioned into <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0005\" wiley:location=\"equation/jgt23212-math-0005.png\"><mrow><mrow><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math> sets <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>V</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <mo>…</mo>\n \n <mo>,</mo>\n \n <msub>\n <mi>V</mi>\n \n <mi>k</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0006\" wiley:location=\"equation/jgt23212-math-0006.png\"><mrow><mrow><msub><mi>V</mi><mn>1</mn></msub><mo>,</mo><mo>\\unicode{x02026}</mo><mo>,</mo><msub><mi>V</mi><mi>k</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> such that every edge in <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>E</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0007\" wiley:location=\"equation/jgt23212-math-0007.png\"><mrow><mrow><mi>E</mi></mrow></mrow></math></annotation>\n </semantics></math> contains precisely one vertex from each <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>V</mi>\n \n <mi>i</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0008\" wiley:location=\"equation/jgt23212-math-0008.png\"><mrow><mrow><msub><mi>V</mi><mi>i</mi></msub></mrow></mrow></math></annotation>\n </semantics></math>. We call such a graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0009\" wiley:location=\"equation/jgt23212-math-0009.png\"><mrow><mrow><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math>-balanced if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mo>∣</mo>\n \n <msub>\n <mi>V</mi>\n \n <mi>i</mi>\n </msub>\n \n <mo>∣</mo>\n \n <mo>=</mo>\n \n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0010\" wiley:location=\"equation/jgt23212-math-0010.png\"><mrow><mrow><mo>\\unicode{x02223}</mo><msub><mi>V</mi><mi>i</mi></msub><mo>\\unicode{x02223}</mo><mo>=</mo><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math> for each <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>i</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0011\" wiley:location=\"equation/jgt23212-math-0011.png\"><mrow><mrow><mi>i</mi></mrow></mrow></math></annotation>\n </semantics></math>. An independent set <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>I</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0012\" wiley:location=\"equation/jgt23212-math-0012.png\"><mrow><mrow><mi>I</mi></mrow></mrow></math></annotation>\n </semantics></math> in <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0013\" wiley:location=\"equation/jgt23212-math-0013.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> is balanced if <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>I</mi>\n \n <mo>∩</mo>\n \n <msub>\n <mi>V</mi>\n \n <mi>i</mi>\n </msub>\n \n <mo>∣</mo>\n \n <mo>=</mo>\n \n <mo>∣</mo>\n \n <mi>I</mi>\n \n <mo>∩</mo>\n \n <msub>\n <mi>V</mi>\n \n <mi>j</mi>\n </msub>\n \n <mo>∣</mo>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0014\" wiley:location=\"equation/jgt23212-math-0014.png\"><mrow><mrow><mo>\\unicode{x02223}</mo><mi>I</mi><mo>\\unicode{x02229}</mo><msub><mi>V</mi><mi>i</mi></msub><mo>\\unicode{x02223}</mo><mo>=</mo><mo>\\unicode{x02223}</mo><mi>I</mi><mo>\\unicode{x02229}</mo><msub><mi>V</mi><mi>j</mi></msub><mo>\\unicode{x02223}</mo></mrow></mrow></math></annotation>\n </semantics></math> for each <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mn>1</mn>\n \n <mo>⩽</mo>\n \n <mi>i</mi>\n \n <mo>,</mo>\n \n <mi>j</mi>\n \n <mo>⩽</mo>\n \n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0015\" wiley:location=\"equation/jgt23212-math-0015.png\"><mrow><mrow><mn>1</mn><mo>\\unicode{x02A7D}</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>\\unicode{x02A7D}</mo><mi>k</mi></mrow></mrow></math></annotation>\n </semantics></math>, and a coloring is balanced if each color class induces a balanced independent set in <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0016\" wiley:location=\"equation/jgt23212-math-0016.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>. In this paper, we provide a lower bound on the balanced independence number <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>α</mi>\n \n <mi>b</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0017\" wiley:location=\"equation/jgt23212-math-0017.png\"><mrow><mrow><msub><mi>\\unicode{x003B1}</mi><mi>b</mi></msub><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> in terms of the average degree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n \n <mo>=</mo>\n \n <mo>∣</mo>\n \n <mi>E</mi>\n \n <mo>∣</mo>\n \n <mo>/</mo>\n \n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0018\" wiley:location=\"equation/jgt23212-math-0018.png\"><mrow><mrow><mi>D</mi><mo>=</mo><mo>\\unicode{x02223}</mo><mi>E</mi><mo>\\unicode{x02223}</mo><mo>/</mo><mi>n</mi></mrow></mrow></math></annotation>\n </semantics></math>, and an upper bound on the balanced chromatic number <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>χ</mi>\n \n <mi>b</mi>\n </msub>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0019\" wiley:location=\"equation/jgt23212-math-0019.png\"><mrow><mrow><msub><mi>\\unicode{x003C7}</mi><mi>b</mi></msub><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> in terms of the maximum degree <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>Δ</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0020\" wiley:location=\"equation/jgt23212-math-0020.png\"><mrow><mrow><mi mathvariant=\"normal\">\\unicode{x00394}</mi></mrow></mrow></math></annotation>\n </semantics></math>. Our results recover those of recent work of Chakraborti for <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n \n <mo>=</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0021\" wiley:location=\"equation/jgt23212-math-0021.png\"><mrow><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></mrow></math></annotation>\n </semantics></math>.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"109 1","pages":"43-51"},"PeriodicalIF":1.0000,"publicationDate":"2025-01-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23212","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23212","RegionNum":3,"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 $k$ -uniform hypergraph $H = (V, E)$ is $k$ -partite if $V$ can be partitioned into $k$ sets $V_{1}, \dots, V_{k}$ such that every edge in $E$ contains precisely one vertex from each $V_{i}$ . We call such a graph $n$ -balanced if $∣ V_{i} ∣ = n$ for each $i$ . An independent set $I$ in $H$ is balanced if $∣ I \cap V_{i} ∣ = ∣ I \cap V_{j} ∣$ for each $1 ⩽ i, j ⩽ k$ , and a coloring is balanced if each color class induces a balanced independent set in $H$ . In this paper, we provide a lower bound on the balanced independence number $α_{b} (H)$ in terms of the average degree $D = ∣ E ∣ / n$ , and an upper bound on the balanced chromatic number $χ_{b} (H)$ in terms of the maximum degree $Δ$ . Our results recover those of recent work of Chakraborti for $k = 2$ .

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超图的平衡独立集与着色

A k< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0001" wiley:location="equation/jgt23212-math-0001.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></math&gt；-均匀超图H = (V，E) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0002”威利:位置= "方程/ jgt23212 -数学- 0002. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> H< / mi> & lt; mo> = & lt; / mo> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> V< / mi> & lt; mo>, & lt; / mo> & lt; mi> E< / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>is k< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0003" wiley:location="equation/jgt23212-math-0003.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；-partite if V< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0004 .png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></math&gt；可以划分为k&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0005" wiley:location="equation/jgt23212-math-0005.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；集合v1，…，V k <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0006”威利:位置= "方程/ jgt23212 -数学- 0006. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> V< / mi> & lt; mn> 1 & lt; / mn> & lt; / msub> & lt; mo> & lt; / mo> & lt; mo> \ unicode {x02026} & lt; / mo> & lt; mo> & lt; / mo> & lt; msub> & lt; mi> V< / mi> & lt; mi> k< / mi> & lt; / msub> & lt; / mrow> & lt; / mrow> & lt; / math>这样，E< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0007" wiley:location="equation/jgt23212-math-0007.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；只包含每个V的一个顶点&lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0008”威利:位置= "方程/ jgt23212 -数学- 0008. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> V< / mi> & lt; mi> i< / mi> & lt; / msub> & lt; / mrow> & lt; / mrow> & lt; / math>． png”祝辞& lt; mrow> & lt; mrow> & lt; mn> 1 & lt; / mn> & lt; mo> \ unicode {x02A7D} & lt; / mo> & lt; mi> i< / mi> & lt; mo>, & lt; / mo> & lt; mi> j< / mi> & lt; mo> \ unicode {x02A7D} & lt; / mo> & lt; mi> k< / mi> & lt; / mrow> & lt; / mrow> & lt; / math>,如果每个颜色类在H&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0016" wiley:location="equation/jgt23212-math-0016.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></math&gt；。在本文中，我们给出了平衡独立数α b (H)的下界。<math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0017 .png"><mrow>< jgt23212- jgt23212-math-0017.png"><mrow><mrow>< mrow>< \unicode{x003B1}</ jgt23212- jgt23212-math-0017.png"><mrow>< /msub>< /msub>< /msub>< /msub>< /msub>< /msub>< (</mo>< H</mi>< /mrow></mrow></mrow></ mrow></mrow></mrow></ mrow></mrow></mrow></ mrow>&lt；D =∣E∣/ n &ltxmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0018 .png"><mrow>< jgt23212- jgt23212-math-0018.png"><mrow><mrow>< D</ jgt23212- jgt23212-math-0018.png"><mrow>< D</ mrow><mrow> D</ mrow>< mo>=</mo>< /mo><mo>\unicode{x02223}</mo>< /mo>< /mo>< /mo>< /mo> /mrow></mrow></ mrow></mrow></ mrow></mrow></ mrow></mrow></ mrow></mrow></ mrow></math>；,以及平衡色数χ b (H)的上界<math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0019 .png"><mrow>< jgt23212- jgt23212-math-0019.png"><mrow><mrow>< mrow>< \unicode{x003C7}</ msub>< /msub>< /msub>< /msub>< /msub>< (</mo>< H</mi><)</mo></mrow></mrow></mrow></ mrow></mrow></mrow></ mrow></mrow></mrow></ mrow></mrow></mrow></ mrow></mrow></mrow></ mrow></mrow></mrow></ mrow></mrow></mrow></ mrow></mrow></mrow></math&gt；根据最大程度Δ &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0020" wiley:location="equation/jgt23212-math-0020.png"><mrow><mrow><mi mathvariant="normal">\unicode{x00394}</ mrow></mrow></ mrow></mrow></ mrow></mrow></math&gt；。我们的结果恢复了Chakraborti最近对k = 2 <math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23212:jgt23212-math-0021"的研究结果。威利:位置= "方程/ jgt23212 -数学- 0021. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> k< / mi> & lt; mo> = & lt; / mo> & lt; mn> 2 & lt; / mn> & lt; / mrow> & lt; / mrow> & lt; / math>．

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来源期刊

Journal of Graph Theory 数学-数学

CiteScore

1.60

自引率

22.20%

发文量

130

审稿时长

6-12 weeks

期刊介绍： The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .

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