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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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 .