广义图兰问题中图对称论证的稳定性

Pub Date : 2024-07-15 DOI:10.1002/jgt.23151
Dániel Gerbner, Hilal Hama Karim
{"title":"广义图兰问题中图对称论证的稳定性","authors":"Dániel Gerbner,&nbsp;Hilal Hama Karim","doi":"10.1002/jgt.23151","DOIUrl":null,"url":null,"abstract":"<p>Given graphs <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> and <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math>, <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>ex</mtext>\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> $\\text{ex}(n,H,F)$</annotation>\n </semantics></math> denotes the largest number of copies of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> in <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math>-free <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>-vertex graphs. Let <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>χ</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>&lt;</mo>\n \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 \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </mrow>\n <annotation> $\\chi (H)\\lt \\chi (F)=r+1$</annotation>\n </semantics></math>. We say that <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> is <i>F-Turán-stable</i> if the following holds. For any <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ε</mi>\n \n <mo>&gt;</mo>\n \n <mn>0</mn>\n </mrow>\n </mrow>\n <annotation> $\\varepsilon \\gt 0$</annotation>\n </semantics></math> there exists <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>δ</mi>\n \n <mo>&gt;</mo>\n \n <mn>0</mn>\n </mrow>\n </mrow>\n <annotation> $\\delta \\gt 0$</annotation>\n </semantics></math> such that if an <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>n</mi>\n </mrow>\n </mrow>\n <annotation> $n$</annotation>\n </semantics></math>-vertex <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>F</mi>\n </mrow>\n </mrow>\n <annotation> $F$</annotation>\n </semantics></math>-free graph <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> contains at least <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mtext>ex</mtext>\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 \n <mo>−</mo>\n \n <mi>δ</mi>\n \n <msup>\n <mi>n</mi>\n \n <mrow>\n <mo>∣</mo>\n \n <mi>V</mi>\n \n <mrow>\n <mo>(</mo>\n \n <mi>H</mi>\n \n <mo>)</mo>\n </mrow>\n \n <mo>∣</mo>\n </mrow>\n </msup>\n </mrow>\n </mrow>\n <annotation> $\\text{ex}(n,H,F)-\\delta {n}^{| V(H)| }$</annotation>\n </semantics></math> copies of <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math>, then the edit distance of <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 the <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-partite Turán graph is at most <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>ε</mi>\n \n <msup>\n <mi>n</mi>\n \n <mn>2</mn>\n </msup>\n </mrow>\n </mrow>\n <annotation> $\\varepsilon {n}^{2}$</annotation>\n </semantics></math>. We say that <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> is <i>weakly F-Turán-stable</i> if the same holds with the Turán graph replaced by any complete <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math>-partite graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>T</mi>\n </mrow>\n </mrow>\n <annotation> $T$</annotation>\n </semantics></math>. It is known that such stability implies exact results in several cases. We show that complete multipartite graphs with chromatic number at most <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math> are weakly <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>r</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> ${K}_{r+1}$</annotation>\n </semantics></math>-Turán-stable. Partly answering a question of Morrison, Nir, Norin, Rzażewski, and Wesolek positively, we show that for every graph <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math>, if <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>r</mi>\n </mrow>\n </mrow>\n <annotation> $r$</annotation>\n </semantics></math> is large enough, then <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> is <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mrow>\n <mi>r</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> ${K}_{r+1}$</annotation>\n </semantics></math>-Turán-stable. Finally, we prove that if <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>H</mi>\n </mrow>\n </mrow>\n <annotation> $H$</annotation>\n </semantics></math> is bipartite, then it is weakly <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <msub>\n <mi>C</mi>\n \n <mrow>\n <mn>2</mn>\n \n <mi>k</mi>\n \n <mo>+</mo>\n \n <mn>1</mn>\n </mrow>\n </msub>\n </mrow>\n </mrow>\n <annotation> ${C}_{2k+1}$</annotation>\n </semantics></math>-Turán-stable for <span></span><math>\n <semantics>\n <mrow>\n \n <mrow>\n <mi>k</mi>\n </mrow>\n </mrow>\n <annotation> $k$</annotation>\n </semantics></math> large enough.</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Stability from graph symmetrization arguments in generalized Turán problems\",\"authors\":\"Dániel Gerbner,&nbsp;Hilal Hama Karim\",\"doi\":\"10.1002/jgt.23151\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Given graphs <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> and <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>F</mi>\\n </mrow>\\n </mrow>\\n <annotation> $F$</annotation>\\n </semantics></math>, <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mtext>ex</mtext>\\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> $\\\\text{ex}(n,H,F)$</annotation>\\n </semantics></math> denotes the largest number of copies of <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> in <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>F</mi>\\n </mrow>\\n </mrow>\\n <annotation> $F$</annotation>\\n </semantics></math>-free <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math>-vertex graphs. Let <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>χ</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>H</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>&lt;</mo>\\n \\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 \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\chi (H)\\\\lt \\\\chi (F)=r+1$</annotation>\\n </semantics></math>. We say that <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> is <i>F-Turán-stable</i> if the following holds. For any <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>ε</mi>\\n \\n <mo>&gt;</mo>\\n \\n <mn>0</mn>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\varepsilon \\\\gt 0$</annotation>\\n </semantics></math> there exists <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>δ</mi>\\n \\n <mo>&gt;</mo>\\n \\n <mn>0</mn>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\delta \\\\gt 0$</annotation>\\n </semantics></math> such that if an <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>n</mi>\\n </mrow>\\n </mrow>\\n <annotation> $n$</annotation>\\n </semantics></math>-vertex <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>F</mi>\\n </mrow>\\n </mrow>\\n <annotation> $F$</annotation>\\n </semantics></math>-free graph <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> contains at least <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mtext>ex</mtext>\\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 \\n <mo>−</mo>\\n \\n <mi>δ</mi>\\n \\n <msup>\\n <mi>n</mi>\\n \\n <mrow>\\n <mo>∣</mo>\\n \\n <mi>V</mi>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mi>H</mi>\\n \\n <mo>)</mo>\\n </mrow>\\n \\n <mo>∣</mo>\\n </mrow>\\n </msup>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\text{ex}(n,H,F)-\\\\delta {n}^{| V(H)| }$</annotation>\\n </semantics></math> copies of <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math>, then the edit distance of <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 the <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math>-partite Turán graph is at most <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>ε</mi>\\n \\n <msup>\\n <mi>n</mi>\\n \\n <mn>2</mn>\\n </msup>\\n </mrow>\\n </mrow>\\n <annotation> $\\\\varepsilon {n}^{2}$</annotation>\\n </semantics></math>. We say that <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> is <i>weakly F-Turán-stable</i> if the same holds with the Turán graph replaced by any complete <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math>-partite graph <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>T</mi>\\n </mrow>\\n </mrow>\\n <annotation> $T$</annotation>\\n </semantics></math>. It is known that such stability implies exact results in several cases. We show that complete multipartite graphs with chromatic number at most <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math> are weakly <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>K</mi>\\n \\n <mrow>\\n <mi>r</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n </mrow>\\n <annotation> ${K}_{r+1}$</annotation>\\n </semantics></math>-Turán-stable. Partly answering a question of Morrison, Nir, Norin, Rzażewski, and Wesolek positively, we show that for every graph <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math>, if <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>r</mi>\\n </mrow>\\n </mrow>\\n <annotation> $r$</annotation>\\n </semantics></math> is large enough, then <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> is <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>K</mi>\\n \\n <mrow>\\n <mi>r</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n </mrow>\\n <annotation> ${K}_{r+1}$</annotation>\\n </semantics></math>-Turán-stable. Finally, we prove that if <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>H</mi>\\n </mrow>\\n </mrow>\\n <annotation> $H$</annotation>\\n </semantics></math> is bipartite, then it is weakly <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <msub>\\n <mi>C</mi>\\n \\n <mrow>\\n <mn>2</mn>\\n \\n <mi>k</mi>\\n \\n <mo>+</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n </msub>\\n </mrow>\\n </mrow>\\n <annotation> ${C}_{2k+1}$</annotation>\\n </semantics></math>-Turán-stable for <span></span><math>\\n <semantics>\\n <mrow>\\n \\n <mrow>\\n <mi>k</mi>\\n </mrow>\\n </mrow>\\n <annotation> $k$</annotation>\\n </semantics></math> large enough.</p>\",\"PeriodicalId\":0,\"journal\":{\"name\":\"\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.0,\"publicationDate\":\"2024-07-15\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23151\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23151","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0

摘要

给定图 和 ,表示无顶点图中 的最大副本数。如果下面的条件成立,我们就说它是 F-Turán-stable 的。对于任意存在这样的图,如果一个无顶点图至少包含 、 的副本,那么 和 的编辑距离最多为 。如果将图兰图替换为任何完整的-部分图后同样成立,我们就说图兰图具有弱F-图兰稳定性。众所周知,这种稳定性意味着几种情况下的精确结果。我们证明了色度数最多的完整多方图是弱-图兰稳定的。我们部分正面回答了莫里森、尼尔、诺林、拉扎耶夫斯基和韦索莱克的一个问题,证明了对于每个图 ,如果足够大,则是-图兰稳定的。最后,我们证明,如果是双向图,那么对于足够大的图,它是弱-图兰稳定的。
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Stability from graph symmetrization arguments in generalized Turán problems

Given graphs H $H$ and F $F$ , ex ( n , H , F ) $\text{ex}(n,H,F)$ denotes the largest number of copies of H $H$ in F $F$ -free n $n$ -vertex graphs. Let χ ( H ) < χ ( F ) = r + 1 $\chi (H)\lt \chi (F)=r+1$ . We say that H $H$ is F-Turán-stable if the following holds. For any ε > 0 $\varepsilon \gt 0$ there exists δ > 0 $\delta \gt 0$ such that if an n $n$ -vertex F $F$ -free graph G $G$ contains at least ex ( n , H , F ) δ n V ( H ) $\text{ex}(n,H,F)-\delta {n}^{| V(H)| }$ copies of H $H$ , then the edit distance of G $G$ and the r $r$ -partite Turán graph is at most ε n 2 $\varepsilon {n}^{2}$ . We say that H $H$ is weakly F-Turán-stable if the same holds with the Turán graph replaced by any complete r $r$ -partite graph T $T$ . It is known that such stability implies exact results in several cases. We show that complete multipartite graphs with chromatic number at most r $r$ are weakly K r + 1 ${K}_{r+1}$ -Turán-stable. Partly answering a question of Morrison, Nir, Norin, Rzażewski, and Wesolek positively, we show that for every graph H $H$ , if r $r$ is large enough, then H $H$ is K r + 1 ${K}_{r+1}$ -Turán-stable. Finally, we prove that if H $H$ is bipartite, then it is weakly C 2 k + 1 ${C}_{2k+1}$ -Turán-stable for k $k$ large enough.

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