Journal of Combinatorial Theory Series B最新文献_第10页

Extended commonality of paths and cycles via Schur convexity 通过舒尔凸性扩展路径和循环的共性

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-01-17 DOI: 10.1016/j.jctb.2023.12.001

Jang Soo Kim , Joonkyung Lee

{"title":"Extended commonality of paths and cycles via Schur convexity","authors":"Jang Soo Kim , Joonkyung Lee","doi":"10.1016/j.jctb.2023.12.001","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.12.001","url":null,"abstract":"<div>A graph H is common if the number of monochromatic copies of H in a 2-edge-colouring of the complete graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub></math> is asymptotically minimised by the random colouring, or equivalently, <math><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>W</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>−</mo><mi>W</mi><mo>)</mo><mo>≥</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>1</mn><mo>−</mo><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msup></math> holds for every graphon <math><mi>W</mi><mo>:</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></math>, where <math><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mo>.</mo><mo>)</mo></math> denotes the homomorphism density of the graph H. Paths and cycles being common is one of the earliest cornerstones in extremal graph theory, due to Mulholland and Smith (1959), Goodman (1959), and Sidorenko (1989).We prove a graph homomorphism inequality that extends the commonality of paths and cycles. Namely, <math><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mi>W</mi><mo>)</mo><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><mi>H</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>−</mo><mi>W</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><msup><mrow><mo>(</mo><mi>W</mi><mo>)</mo></mrow><mrow><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msup><mo>+</mo><msub><mrow><mi>t</mi></mrow><mrow><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow></msub><msup><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>W</mi><mo>)</mo></mrow><mrow><mi>e</mi><mo>(</mo><mi>H</mi><mo>)</mo></mrow></msup></math> whenever H is a path or a cycle and <math><mi>W</mi><mo>:</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><mi>R</mi></math> is a bounded symmetric measurable function.This answers a question of Sidorenko from 1989, who proved a slightly weaker result for even-length paths to prove the commonality of odd cycles. Furthermore, it also settles a recent conjecture of Behague, Morrison, and Noel in a strong form, who asked if the inequality holds for graphons W and odd cycles H. Our proof uses Schur convexity of complete homogeneous symmetric functions, which may be of independent interest.</div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"166 ","pages":"Pages 109-122"},"PeriodicalIF":1.4,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139487840","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A critical probability for biclique partition of Gn,p Gn 的双斜分区临界概率 p

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-01-12 DOI: 10.1016/j.jctb.2023.12.005

Tom Bohman , Jakob Hofstad

引用次数: 0

Excluded minors for the Klein bottle II. Cascades 克莱因瓶 II 不包括未成年人。级联

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-01-12 DOI: 10.1016/j.jctb.2023.12.006

Bojan Mohar , Petr Škoda

引用次数: 0

Sparse graphs without long induced paths 没有长诱导路径的稀疏图

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-01-05 DOI: 10.1016/j.jctb.2023.12.003

Oscar Defrain , Jean-Florent Raymond

引用次数: 0

Count and cofactor matroids of highly connected graphs 高连接图的计数和共因矩阵

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2024-01-05 DOI: 10.1016/j.jctb.2023.12.004

Dániel Garamvölgyi , Tibor Jordán , Csaba Király

引用次数: 0

Turán graphs with bounded matching number 匹配数有界的图兰图

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2023-12-15 DOI: 10.1016/j.jctb.2023.12.002

Noga Alon , Péter Frankl

引用次数: 0

On a problem of El-Zahar and Erdős 关于扎哈尔和厄尔多斯的一个问题

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2023-12-11 DOI: 10.1016/j.jctb.2023.11.004

Tung Nguyen , Alex Scott , Paul Seymour

{"title":"On a problem of El-Zahar and Erdős","authors":"Tung Nguyen , Alex Scott , Paul Seymour","doi":"10.1016/j.jctb.2023.11.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.11.004","url":null,"abstract":"<div>Two subgraphs <math><mi>A</mi><mo>,</mo><mi>B</mi></math> of a graph G are anticomplete if they are vertex-disjoint and there are no edges joining them. Is it true that if G is a graph with bounded clique number, and sufficiently large chromatic number, then it has two anticomplete subgraphs, both with large chromatic number? This is a question raised by El-Zahar and Erdős in 1986, and remains open. If so, then at least there should be two anticomplete subgraphs both with large minimum degree, and that is one of our results.We prove two variants of this. First, a strengthening: we can ask for one of the two subgraphs to have large chromatic number: that is, for all <math><mi>t</mi><mo>,</mo><mi>c</mi><mo>≥</mo><mn>1</mn></math> there exists <math><mi>d</mi><mo>≥</mo><mn>1</mn></math> such that if G has chromatic number at least d, and does not contain the complete graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math> as a subgraph, then there are anticomplete subgraphs <math><mi>A</mi><mo>,</mo><mi>B</mi></math>, where A has minimum degree at least c and B has chromatic number at least c.Second, we look at what happens if we replace the hypothesis that G has sufficiently large chromatic number with the hypothesis that G has sufficiently large minimum degree. This, together with excluding <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math>, is not enough to guarantee two anticomplete subgraphs both with large minimum degree; but it works if instead of excluding <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math> we exclude the complete bipartite graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math>. More exactly: for all <math><mi>t</mi><mo>,</mo><mi>c</mi><mo>≥</mo><mn>1</mn></math> there exists <math><mi>d</mi><mo>≥</mo><mn>1</mn></math> such that if G has minimum degree at least d, and does not contain the complete bipartite graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math> as a subgraph, then there are two anticomplete subgraphs both with minimum degree at least c.</div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"165 ","pages":"Pages 211-222"},"PeriodicalIF":1.4,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138570098","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Graph partitions under average degree constraint 平均度约束下的图分区

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2023-12-05 DOI: 10.1016/j.jctb.2023.11.006

Yan Wang , Hehui Wu

引用次数: 0

Hitting all maximum stable sets in P5-free graphs 在P5-free图中命中所有最大稳定集

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2023-11-29 DOI: 10.1016/j.jctb.2023.11.005

Sepehr Hajebi , Yanjia Li , Sophie Spirkl

{"title":"Hitting all maximum stable sets in P5-free graphs","authors":"Sepehr Hajebi , Yanjia Li , Sophie Spirkl","doi":"10.1016/j.jctb.2023.11.005","DOIUrl":"10.1016/j.jctb.2023.11.005","url":null,"abstract":"<div>We prove that every <math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math>-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where <math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math> denotes the t-vertex path, and for graphs <math><mi>G</mi><mo>,</mo><mi>H</mi></math>, we say G is H-free if no induced subgraph of G is isomorphic to H).More generally, let us say a class <math><mi>C</mi></math> of graphs is η-bounded if there exists a function <math><mi>h</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math> such that <math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mi>h</mi><mo>(</mo><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>)</mo></math> for every graph <math><mi>G</mi><mo>∈</mo><mi>C</mi></math>, where <math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> denotes smallest cardinality of a hitting set of all maximum stable sets in G, and <math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> is the clique number of G. Also, <math><mi>C</mi></math> is said to be polynomially η-bounded if in addition h can be chosen to be a polynomial.We introduce η-boundedness inspired by a question of Alon (asking how large <math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> can be for a 3-colourable graph G), and motivated by a number of meaningful similarities to χ-boundedness, namely,<ul><li>•given a graph G, we have <math><mi>η</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>ω</mi><mo>(</mo><mi>H</mi><mo>)</mo></math> for every induced subgraph H of G if and only if G is perfect;</li><li>•there are graphs G with both <math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math> and the girth of G arbitrarily large; and</li><li>•if <math><mi>C</mi></math> is a hereditary class of graphs which is polynomially η-bounded, then <math><mi>C</mi></math> satisfies the Erdős-Hajnal conjecture.</li></ul> The second bullet above in particular suggests an analogue of the Gyárfás-Sumner conjecture, that the class of all H-free graphs is η-bounded if (and only if) H is a forest. Like χ-boundedness, the case where H is a star is easy to verify, and we prove two non-trivial extensions of this: H-free graphs are η-bounded if (1) H has a vertex incident with all edges of H, or (2) H can be obtained from a star by subdividing at most one edge, exactly once.Unlike χ-boundedness","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":"165 ","pages":"Pages 142-163"},"PeriodicalIF":1.4,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138455110","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 3

Dimension is polynomial in height for posets with planar cover graphs 对于具有平面覆盖图的偏置集，维度是高度的多项式

IF 1.4 1区数学

Journal of Combinatorial Theory Series B Pub Date : 2023-11-29 DOI: 10.1016/j.jctb.2023.10.009

Jakub Kozik , Piotr Micek , William T. Trotter

引用次数: 8