{"title":"Cliques in squares of graphs with maximum average degree less than 4","authors":"Daniel W. Cranston, Gexin Yu","doi":"10.1002/jgt.23125","DOIUrl":null,"url":null,"abstract":"<p>Hocquard, Kim, and Pierron constructed, for every even integer <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n \n <mo>≥</mo>\n \n <mn>2</mn>\n </mrow>\n </mrow>\n <annotation> $D\\ge 2$</annotation>\n </semantics></math>, a 2-degenerate graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mi>D</mi>\n </msub>\n </mrow>\n </mrow>\n <annotation> ${G}_{D}$</annotation>\n </semantics></math> with maximum degree <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n </mrow>\n </mrow>\n <annotation> $D$</annotation>\n </semantics></math> such that <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msubsup>\n <mi>G</mi>\n \n <mi>D</mi>\n \n <mn>2</mn>\n </msubsup>\n \n <mo>)</mo>\n </mrow>\n \n <mo>=</mo>\n \n <mfrac>\n <mn>5</mn>\n \n <mn>2</mn>\n </mfrac>\n \n <mi>D</mi>\n </mrow>\n </mrow>\n <annotation> $\\omega ({G}_{D}^{2})=\\frac{5}{2}D$</annotation>\n </semantics></math>. We prove for (a) all 2-degenerate graphs <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> and (b) all graphs <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> with <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>mad</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo><</mo>\n \n <mn>4</mn>\n </mrow>\n </mrow>\n <annotation> $\\text{mad}(G)\\lt 4$</annotation>\n </semantics></math>, upper bounds on the clique number <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>G</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mrow>\n <annotation> $\\omega ({G}^{2})$</annotation>\n </semantics></math> of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>G</mi>\n \n <mn>2</mn>\n </msup>\n </mrow>\n </mrow>\n <annotation> ${G}^{2}$</annotation>\n </semantics></math> that match the lower bound given by this construction, up to small additive constants. We show that if <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> is 2-degenerate with maximum degree <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n </mrow>\n </mrow>\n <annotation> $D$</annotation>\n </semantics></math>, then <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>G</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mfrac>\n <mn>5</mn>\n \n <mn>2</mn>\n </mfrac>\n \n <mi>D</mi>\n \n <mo>+</mo>\n \n <mn>72</mn>\n </mrow>\n </mrow>\n <annotation> $\\omega ({G}^{2})\\le \\frac{5}{2}D+72$</annotation>\n </semantics></math> (with <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>G</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mfrac>\n <mn>5</mn>\n \n <mn>2</mn>\n </mfrac>\n \n <mi>D</mi>\n \n <mo>+</mo>\n \n <mn>60</mn>\n </mrow>\n </mrow>\n <annotation> $\\omega ({G}^{2})\\le \\frac{5}{2}D+60$</annotation>\n </semantics></math> when <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n </mrow>\n </mrow>\n <annotation> $D$</annotation>\n </semantics></math> is sufficiently large). And if <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> has <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>mad</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mi>G</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo><</mo>\n \n <mn>4</mn>\n </mrow>\n </mrow>\n <annotation> $\\text{mad}(G)\\lt 4$</annotation>\n </semantics></math> and maximum degree <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>D</mi>\n </mrow>\n </mrow>\n <annotation> $D$</annotation>\n </semantics></math>, then <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ω</mi>\n \n <mrow>\n <mo>(</mo>\n \n <msup>\n <mi>G</mi>\n \n <mn>2</mn>\n </msup>\n \n <mo>)</mo>\n </mrow>\n \n <mo>≤</mo>\n \n <mfrac>\n <mn>5</mn>\n \n <mn>2</mn>\n </mfrac>\n \n <mi>D</mi>\n \n <mo>+</mo>\n \n <mn>532</mn>\n </mrow>\n </mrow>\n <annotation> $\\omega ({G}^{2})\\le \\frac{5}{2}D+532$</annotation>\n </semantics></math>. Thus, the construction of Hocquard et al. is essentially the best possible. Our proofs introduce a “token passing” technique to derive crucial information about nonadjacencies in <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>G</mi>\n </mrow>\n </mrow>\n <annotation> $G$</annotation>\n </semantics></math> of vertices that are adjacent in <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msup>\n <mi>G</mi>\n \n <mn>2</mn>\n </msup>\n </mrow>\n </mrow>\n <annotation> ${G}^{2}$</annotation>\n </semantics></math>. This is a powerful technique for working with such graphs that has not previously appeared in the literature.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"107 3","pages":"559-577"},"PeriodicalIF":0.9000,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23125","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23125","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
Hocquard, Kim, and Pierron constructed, for every even integer , a 2-degenerate graph with maximum degree such that . We prove for (a) all 2-degenerate graphs and (b) all graphs with , upper bounds on the clique number of that match the lower bound given by this construction, up to small additive constants. We show that if is 2-degenerate with maximum degree , then (with when is sufficiently large). And if has and maximum degree , then . Thus, the construction of Hocquard et al. is essentially the best possible. Our proofs introduce a “token passing” technique to derive crucial information about nonadjacencies in of vertices that are adjacent in . This is a powerful technique for working with such graphs that has not previously appeared in the literature.
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.
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