On the Constructor–Blocker game

IF 0.9 3区数学 Q2 MATHEMATICS

Journal of Graph Theory Pub Date : 2024-10-06 DOI:10.1002/jgt.23186

József Balogh, Ce Chen, Sean English

{"title":"On the Constructor–Blocker game","authors":"József Balogh, Ce Chen, Sean English","doi":"10.1002/jgt.23186","DOIUrl":null,"url":null,"abstract":"In the Constructor–Blocker game, two players, Constructor and Blocker, alternately claim unclaimed edges of the complete graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mi>n</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0001\" wiley:location=\"equation/jgt23186-math-0001.png\"><mrow><mrow><msub><mi>K</mi><mi>n</mi></msub></mrow></mrow></math></annotation>\n </semantics></math>. For given graphs <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0002\" wiley:location=\"equation/jgt23186-math-0002.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math> and <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:jgt23186:jgt23186-math-0003\" wiley:location=\"equation/jgt23186-math-0003.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math>, Constructor can only claim edges that leave her graph <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0004\" wiley:location=\"equation/jgt23186-math-0004.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math>-free, while Blocker has no restrictions. Constructor's goal is to build as many copies of <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:jgt23186:jgt23186-math-0005\" wiley:location=\"equation/jgt23186-math-0005.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> as she can, while Blocker attempts to minimize the number of copies of <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:jgt23186:jgt23186-math-0006\" wiley:location=\"equation/jgt23186-math-0006.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> in Constructor's graph. The game ends once there are no more edges that Constructor can claim. The score <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>g</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>H</mi>\n \n <mo>,</mo>\n \n <mi>F</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:jgt23186:jgt23186-math-0007\" wiley:location=\"equation/jgt23186-math-0007.png\"><mrow><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> of the game is the number of copies of <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:jgt23186:jgt23186-math-0008\" wiley:location=\"equation/jgt23186-math-0008.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> in Constructor's graph at the end of the game when both players play optimally and Constructor plays first. In this paper, we extend results of Patkós, Stojaković and Vizer on <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>g</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>H</mi>\n \n <mo>,</mo>\n \n <mi>F</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:jgt23186:jgt23186-math-0009\" wiley:location=\"equation/jgt23186-math-0009.png\"><mrow><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> to many pairs of <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:jgt23186:jgt23186-math-0010\" wiley:location=\"equation/jgt23186-math-0010.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0011\" wiley:location=\"equation/jgt23186-math-0011.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math>: We determine <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>g</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>H</mi>\n \n <mo>,</mo>\n \n <mi>F</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:jgt23186:jgt23186-math-0012\" wiley:location=\"equation/jgt23186-math-0012.png\"><mrow><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> when <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n \n <mo>=</mo>\n \n <msub>\n <mi>K</mi>\n \n <mi>r</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0013\" wiley:location=\"equation/jgt23186-math-0013.png\"><mrow><mrow><mi>H</mi><mo>=</mo><msub><mi>K</mi><mi>r</mi></msub></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>F</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>></mo>\n \n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0014\" wiley:location=\"equation/jgt23186-math-0014.png\"><mrow><mrow><mi>\\unicode{x003C7}</mi><mrow><mo>(</mo><mi>F</mi><mo>)</mo></mrow><mo>\\unicode{x0003E}</mo><mi>r</mi></mrow></mrow></math></annotation>\n </semantics></math>, also when both <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:jgt23186:jgt23186-math-0015\" wiley:location=\"equation/jgt23186-math-0015.png\"><mrow><mrow><mi>H</mi></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0016\" wiley:location=\"equation/jgt23186-math-0016.png\"><mrow><mrow><mi>F</mi></mrow></mrow></math></annotation>\n </semantics></math> are odd cycles, using Szemerédi's Regularity Lemma. We also obtain bounds of <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>g</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <mi>n</mi>\n \n <mo>,</mo>\n \n <mi>H</mi>\n \n <mo>,</mo>\n \n <mi>F</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:jgt23186:jgt23186-math-0017\" wiley:location=\"equation/jgt23186-math-0017.png\"><mrow><mrow><mi>g</mi><mrow><mo>(</mo><mrow><mi>n</mi><mo>,</mo><mi>H</mi><mo>,</mo><mi>F</mi></mrow><mo>)</mo></mrow></mrow></mrow></math></annotation>\n </semantics></math> when <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n \n <mo>=</mo>\n \n <msub>\n <mi>K</mi>\n \n <mn>3</mn>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0018\" wiley:location=\"equation/jgt23186-math-0018.png\"><mrow><mrow><mi>H</mi><mo>=</mo><msub><mi>K</mi><mn>3</mn></msub></mrow></mrow></math></annotation>\n </semantics></math> and <math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n \n <mo>=</mo>\n \n <msub>\n <mi>K</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mo>,</mo>\n \n <mn>2</mn>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> <math xmlns=\"http://www.w3.org/1998/Math/MathML\" altimg=\"urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0019\" wiley:location=\"equation/jgt23186-math-0019.png\"><mrow><mrow><mi>F</mi><mo>=</mo><msub><mi>K</mi><mrow><mn>2</mn><mo>,</mo><mn>2</mn></mrow></msub></mrow></mrow></math></annotation>\n </semantics></math>.","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"108 3","pages":"492-507"},"PeriodicalIF":0.9000,"publicationDate":"2024-10-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23186","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

In the Constructor–Blocker game, two players, Constructor and Blocker, alternately claim unclaimed edges of the complete graph $K_{n}$ . For given graphs $F$ and $H$ , Constructor can only claim edges that leave her graph $F$ -free, while Blocker has no restrictions. Constructor's goal is to build as many copies of $H$ as she can, while Blocker attempts to minimize the number of copies of $H$ in Constructor's graph. The game ends once there are no more edges that Constructor can claim. The score $g (n, H, F)$ of the game is the number of copies of $H$ in Constructor's graph at the end of the game when both players play optimally and Constructor plays first. In this paper, we extend results of Patkós, Stojaković and Vizer on $g (n, H, F)$ to many pairs of $H$ and $F$ : We determine $g (n, H, F)$ when $H = K_{r}$ and $χ (F) > r$ , also when both $H$ and $F$ are odd cycles, using Szemerédi's Regularity Lemma. We also obtain bounds of $g (n, H, F)$ when $H = K_{3}$ and $F = K_{2, 2}$ .

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在建造者-拦阻者游戏中

在建造者-拦阻者游戏中，两个玩家，建造者和拦阻者，交替声明完全图的未声明边K n &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0001”威利:位置= "方程/ jgt23186 -数学- 0001. png”祝辞& lt; mrow> & lt; mrow> & lt; msub> & lt; mi> K< / mi> & lt; mi> n< / mi> & lt; / msub> & lt; / mrow> & lt; / mrow> & lt; / math>．对于给定的图形F&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0002" wiley:location="equation/jgt23186-math-0002.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；and H< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0003" wiley:location="equation/jgt23186-math-0003.png"><mrow><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；， Constructor只能声明离开她的图的边F&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0004" wiley:location="equation/jgt23186-math-0004.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；免费，而Blocker没有限制。构造器的目标是构建尽可能多的H&lt； math的副本xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0005" wiley:location="equation/jgt23186-math-0005.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；当Blocker尝试尽量减少H&lt； math的副本数量时，xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0006" wiley:location="equation/jgt23186-math-0006.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></mrow></math&gt；在构造函数的图中。游戏结束时，没有更多的边构造函数可以要求。分数g (n， H，F) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0007”威利:位置= "方程/ jgt23186 -数学- 0007. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> g< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; mi> H< / mi> & lt; mo>, & lt; / mo> & lt; mi> F< / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>游戏的H&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0008" wiley:location="equation/jgt23186-math-0008.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math>；在游戏结束时，两个玩家都处于最佳状态，而构造函数先玩。本文推广了Patkós、stojakoviki和Vizer关于g (n， H，F) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0009”威利:位置= "方程/ jgt23186 -数学- 0009. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> g< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; mi> H< / mi> & lt; mo>, & lt; / mo> & lt; mi> F< / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>多对H&lt； math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0010" wiley:location="equation/jgt23186-math-0010.png"><mrow>< /mrow></mrow></ mrow></mrow></ mrow></math&gt；and F< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0011" wiley:location="equation/jgt23186-math-0011.png"><mrow>< /mrow></mrow></ mrow></mrow>&lt；:我们确定g (n， H，F) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0012”威利:位置= "方程/ jgt23186 -数学- 0012. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> g< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; mi> H< / mi> & lt; mo>, & lt; / mo> & lt; mi> F< / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>当H = K r <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0013”威利:位置= "方程/ jgt23186 -数学- 0013. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> H< / mi> & lt; mo> = & lt; / mo> & lt; msub> & lt; mi> K< / mi> & lt; mi> r< / mi> & lt; / msub> & lt; / mrow> & lt; / mrow> & lt; / math>和χ (F) &gt；r< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186- jgt23186-math-0014.png"><mrow><mrow>< " wiley:location="equation/jgt23186-math-0014.png"><mrow><mrow>< \unicode{x003C7}</ jgt23186; jgt23186- jgt23186- jgt23186- jgt23186- jgt23186- jgt23186- jgt23186- jgt23186- jgt23186- mrow>< \unicode{x003C7}</ mrow>< (</ mrow><mrow>< mrow>< (</ mrow><) mrow>< (</ mrow><) mrow>< (</ mrow><) mrow>< (mrow><) mrow>< (mrow><) mrow></mrow></ mrow></mrow></ mrow>&lt；，也当H &lt；math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0015" wiley:location="equation/jgt23186-math-0015。 png”祝辞& lt; mrow> & lt; mrow> & lt; mi> H< / mi> & lt; / mrow> & lt; / mrow> & lt; / math>and F< math xmlns="http://www.w3.org/1998/Math/MathML" altimg="urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0016" wiley:location="equation/jgt23186-math-0016.png"><mrow>< /mrow></mrow></ mrow></mrow>&lt；是奇循环，利用szemersamedi的正则引理。我们也得到了g (n， H，F) <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0017”威利:位置= "方程/ jgt23186 -数学- 0017. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> g< / mi> & lt; mrow> & lt; mo> (& lt; / mo> & lt; mrow> & lt; mi> n< / mi> & lt; mo>, & lt; / mo> & lt; mi> H< / mi> & lt; mo>, & lt; / mo> & lt; mi> F< / mi> & lt; / mrow> & lt; mo>) & lt; / mo> & lt; / mrow> & lt; / mrow> & lt; / mrow> & lt; / math>当H = K 3 <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0018”威利:位置= "方程/ jgt23186 -数学- 0018. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> H< / mi> & lt; mo> = & lt; / mo> & lt; msub> & lt; mi> K< / mi> & lt; mn> 3 & lt; / mn> & lt; / msub> & lt; / mrow> & lt; / mrow> & lt; / math>F = k2，2 <math xmlns="http://www.w3.org/1998/Math/MathML" altimg=“urn:x-wiley:03649024:media:jgt23186:jgt23186-math-0019”威利:位置= "方程/ jgt23186 -数学- 0019. png”祝辞& lt; mrow> & lt; mrow> & lt; mi> F< / mi> & lt; mo> = & lt; / mo> & lt; msub> & lt; mi> K< / mi> & lt; mrow> & lt; mn> 2 & lt; / mn> & lt; mo> & lt; / mo> & lt; mn> 2 & lt; / mn> & lt; / mrow> & lt; / msub> & 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 .