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List-k-Coloring H-Free Graphs for All $$k>4$$ 针对所有 $$k>4$$ 的列表-k-着色无 H 图
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-05-14 DOI: 10.1007/s00493-024-00106-2
Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl
{"title":"List-k-Coloring H-Free Graphs for All $$k>4$$","authors":"Maria Chudnovsky, Sepehr Hajebi, Sophie Spirkl","doi":"10.1007/s00493-024-00106-2","DOIUrl":"https://doi.org/10.1007/s00493-024-00106-2","url":null,"abstract":"<p>Given an integer <span>(k&gt;4)</span> and a graph <i>H</i>, we prove that, assuming <span>P</span><span>(ne )</span> <span>NP</span>, the <span>List-</span><i>k</i> <span>-Coloring Problem</span> restricted to <i>H</i>-free graphs can be solved in polynomial time if and only if either every component of <i>H</i> is a path on at most three vertices, or removing the isolated vertices of <i>H</i> leaves an induced subgraph of the five-vertex path. In fact, the “if” implication holds for all <span>(kge 1)</span>.\u0000</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"47 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140942758","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Rainbow Cycles in Properly Edge-Colored Graphs 适当边缘着色图形中的彩虹循环
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-05-02 DOI: 10.1007/s00493-024-00101-7
Jaehoon Kim, Joonkyung Lee, Hong Liu, Tuan Tran
{"title":"Rainbow Cycles in Properly Edge-Colored Graphs","authors":"Jaehoon Kim, Joonkyung Lee, Hong Liu, Tuan Tran","doi":"10.1007/s00493-024-00101-7","DOIUrl":"https://doi.org/10.1007/s00493-024-00101-7","url":null,"abstract":"<p>We prove that every properly edge-colored <i>n</i>-vertex graph with average degree at least <span>(32(log 5n)^2)</span> contains a rainbow cycle, improving upon the <span>((log n)^{2+o(1)})</span> bound due to Tomon. We also prove that every properly edge-colored <i>n</i>-vertex graph with at least <span>(10^5 k^3 n^{1+1/k})</span> edges contains a rainbow 2<i>k</i>-cycle, which improves the previous bound <span>(2^{ck^2}n^{1+1/k})</span> obtained by Janzer. Our method using homomorphism inequalities and a lopsided regularization lemma also provides a simple way to prove the Erdős–Simonovits supersaturation theorem for even cycles, which may be of independent interest.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"58 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140819527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Rainbow Variations on a Theme by Mantel: Extremal Problems for Gallai Colouring Templates 曼特尔的主题彩虹变奏曲：加莱填色模板的极值问题
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-04-29 DOI: 10.1007/s00493-024-00102-6
Victor Falgas-Ravry, Klas Markström, Eero Räty
{"title":"Rainbow Variations on a Theme by Mantel: Extremal Problems for Gallai Colouring Templates","authors":"Victor Falgas-Ravry, Klas Markström, Eero Räty","doi":"10.1007/s00493-024-00102-6","DOIUrl":"https://doi.org/10.1007/s00493-024-00102-6","url":null,"abstract":"<p>Let <span>(textbf{G}:=(G_1, G_2, G_3))</span> be a triple of graphs on the same vertex set <i>V</i> of size <i>n</i>. A rainbow triangle in <span>(textbf{G})</span> is a triple of edges <span>((e_1, e_2, e_3))</span> with <span>(e_iin G_i)</span> for each <i>i</i> and <span>({e_1, e_2, e_3})</span> forming a triangle in <i>V</i>. The triples <span>(textbf{G})</span> not containing rainbow triangles, also known as Gallai colouring templates, are a widely studied class of objects in extremal combinatorics. In the present work, we fully determine the set of edge densities <span>((alpha _1, alpha _2, alpha _3))</span> such that if <span>(vert E(G_i)vert &gt; alpha _i n^2)</span> for each <i>i</i> and <i>n</i> is sufficiently large, then <span>(textbf{G})</span> must contain a rainbow triangle. This resolves a problem raised by Aharoni, DeVos, de la Maza, Montejanos and Šámal, generalises several previous results on extremal Gallai colouring templates, and proves a recent conjecture of Frankl, Győri, He, Lv, Salia, Tompkins, Varga and Zhu.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"70 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140808486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Lower Bound for Essential Covers of the Cube 立方体基本封面的下限
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-04-23 DOI: 10.1007/s00493-024-00093-4
Gal Yehuda, Amir Yehudayoff
{"title":"A Lower Bound for Essential Covers of the Cube","authors":"Gal Yehuda, Amir Yehudayoff","doi":"10.1007/s00493-024-00093-4","DOIUrl":"https://doi.org/10.1007/s00493-024-00093-4","url":null,"abstract":"<p>The amount of hyperplanes that are needed in order to cover the Boolean cube has been studied in various contexts. Linial and Radhakrishnan introduced the notion of <i>essential</i> covers. An essential cover is a collection of hyperplanes that form a minimal cover of the vertices of the hypercube, and every coordinate is influential in at least one of the hyperplanes. Linial and Radhakrishnan proved using algebraic tools that every essential cover of the <i>n</i>-cube must be of size at least <span>(Omega (sqrt{n}))</span>. We devise a stronger lower bound method, and show that the size of every essential cover is at least <span>(Omega (n^{0.52}))</span>. This result has implications in proof complexity, because essential covers have been used to prove lower bounds for several proof systems.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"138 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636086","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Note on the Theorem of Balog, Szemerédi, and Gowers 关于巴洛格、塞梅雷迪和高尔斯定理的说明
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-04-23 DOI: 10.1007/s00493-024-00092-5
Christian Reiher, Tomasz Schoen
{"title":"Note on the Theorem of Balog, Szemerédi, and Gowers","authors":"Christian Reiher, Tomasz Schoen","doi":"10.1007/s00493-024-00092-5","DOIUrl":"https://doi.org/10.1007/s00493-024-00092-5","url":null,"abstract":"<p>We prove that every additive set <i>A</i> with energy <span>(E(A)ge |A|^3/K)</span> has a subset <span>(A'subseteq A)</span> of size <span>(|A'|ge (1-varepsilon )K^{-1/2}|A|)</span> such that <span>(|A'-A'|le O_varepsilon (K^{4}|A'|))</span>. This is, essentially, the largest structured set one can get in the Balog–Szemerédi–Gowers theorem.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"53 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140636316","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Bounding the Chromatic Number of Dense Digraphs by Arc Neighborhoods 通过弧邻域限定密集图谱的色度数
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-04-17 DOI: 10.1007/s00493-024-00098-z
Felix Klingelhoefer, Alantha Newman
{"title":"Bounding the Chromatic Number of Dense Digraphs by Arc Neighborhoods","authors":"Felix Klingelhoefer, Alantha Newman","doi":"10.1007/s00493-024-00098-z","DOIUrl":"https://doi.org/10.1007/s00493-024-00098-z","url":null,"abstract":"<p>The chromatic number of a directed graph is the minimum number of induced acyclic subdigraphs that cover its vertex set, and accordingly, the chromatic number of a tournament is the minimum number of transitive subtournaments that cover its vertex set. The neighborhood of an arc <i>uv</i> in a tournament <i>T</i> is the set of vertices that form a directed triangle with arc <i>uv</i>. We show that if the neighborhood of every arc in a tournament has bounded chromatic number, then the whole tournament has bounded chromatic number. This holds more generally for oriented graphs with bounded independence number, and we extend our proof from tournaments to this class of dense digraphs. As an application, we prove the equivalence of a conjecture of El-Zahar and Erdős and a recent conjecture of Nguyen, Scott and Seymour relating the structure of graphs and tournaments with high chromatic number.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"60 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603585","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Solution to a Problem of Grünbaum on the Edge Density of 4-Critical Planar Graphs 格伦鲍姆关于四临界平面图边缘密度问题的解答
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-04-17 DOI: 10.1007/s00493-024-00100-8
Zdeněk Dvořák, Carl Feghali
{"title":"Solution to a Problem of Grünbaum on the Edge Density of 4-Critical Planar Graphs","authors":"Zdeněk Dvořák, Carl Feghali","doi":"10.1007/s00493-024-00100-8","DOIUrl":"https://doi.org/10.1007/s00493-024-00100-8","url":null,"abstract":"<p>We show that <span>(limsup |E(G)|/|V(G)| = 2.5)</span> over all 4-critical planar graphs <i>G</i>, answering a question of Grünbaum from 1988.\u0000</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"74 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603532","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Turán Density of Long Tight Cycle Minus One Hyperedge 减去一个海波里奇的长密周期的图兰密度
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-04-17 DOI: 10.1007/s00493-024-00099-y
József Balogh, Haoran Luo
{"title":"Turán Density of Long Tight Cycle Minus One Hyperedge","authors":"József Balogh, Haoran Luo","doi":"10.1007/s00493-024-00099-y","DOIUrl":"https://doi.org/10.1007/s00493-024-00099-y","url":null,"abstract":"<p>Denote by <span>({mathcal {C}}^-_{ell })</span> the 3-uniform hypergraph obtained by removing one hyperedge from the tight cycle on <span>(ell )</span> vertices. It is conjectured that the Turán density of <span>({mathcal {C}}^-_{5})</span> is 1/4. In this paper, we make progress toward this conjecture by proving that the Turán density of <span>({mathcal {C}}^-_{ell })</span> is 1/4, for every sufficiently large <span>(ell )</span> not divisible by 3. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament. A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidický.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"301 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603527","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Small Subgraphs with Large Average Degree 平均度大的小子图
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-04-15 DOI: 10.1007/s00493-024-00091-6
Oliver Janzer, Benny Sudakov, István Tomon
{"title":"Small Subgraphs with Large Average Degree","authors":"Oliver Janzer, Benny Sudakov, István Tomon","doi":"10.1007/s00493-024-00091-6","DOIUrl":"https://doi.org/10.1007/s00493-024-00091-6","url":null,"abstract":"<p>In this paper we study the fundamental problem of finding small dense subgraphs in a given graph. For a real number <span>(s&gt;2)</span>, we prove that every graph on <i>n</i> vertices with average degree <span>(dge s)</span> contains a subgraph of average degree at least <i>s</i> on at most <span>(nd^{-frac{s}{s-2}}(log d)^{O_s(1)})</span> vertices. This is optimal up to the polylogarithmic factor, and resolves a conjecture of Feige and Wagner. In addition, we show that every graph with <i>n</i> vertices and average degree at least <span>(n^{1-frac{2}{s}+varepsilon })</span> contains a subgraph of average degree at least <i>s</i> on <span>(O_{varepsilon ,s}(1))</span> vertices, which is also optimal up to the constant hidden in the <i>O</i>(.) notation, and resolves a conjecture of Verstraëte.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"55 9 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140553221","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Hypergraph Analog of Dirac’s Theorem for Long Cycles in 2-Connected Graphs 二连图中长循环的狄拉克超图相似定理
IF 1.1 2区 数学
Combinatorica Pub Date : 2024-04-15 DOI: 10.1007/s00493-024-00096-1
Alexandr Kostochka, Ruth Luo, Grace McCourt
{"title":"A Hypergraph Analog of Dirac’s Theorem for Long Cycles in 2-Connected Graphs","authors":"Alexandr Kostochka, Ruth Luo, Grace McCourt","doi":"10.1007/s00493-024-00096-1","DOIUrl":"https://doi.org/10.1007/s00493-024-00096-1","url":null,"abstract":"<p>Dirac proved that each <i>n</i>-vertex 2-connected graph with minimum degree at least <i>k</i> contains a cycle of length at least <span>(min {2k, n})</span>. We consider a hypergraph version of this result. A <i>Berge cycle</i> in a hypergraph is an alternating sequence of distinct vertices and edges <span>(v_1,e_2,v_2, ldots , e_c, v_1)</span> such that <span>({v_i,v_{i+1}} subseteq e_i)</span> for all <i>i</i> (with indices taken modulo <i>c</i>). We prove that for <span>(n ge k ge r+2 ge 5)</span>, every 2-connected <i>r</i>-uniform <i>n</i>-vertex hypergraph with minimum degree at least <span>({k-1 atopwithdelims ()r-1} + 1)</span> has a Berge cycle of length at least <span>(min {2k, n})</span>. The bound is exact for all <span>(kge r+2ge 5)</span>.</p>","PeriodicalId":50666,"journal":{"name":"Combinatorica","volume":"27 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140553253","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
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