Pub Date : 2024-01-12DOI: 10.1016/j.jctb.2023.12.006
Bojan Mohar , Petr Škoda
Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus k or for embeddings into nonorientable surface of genus k are built from 3-connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the Klein bottle is obtained.
研究了对嵌入曲面至关重要的图形（最小排除最小）。在第一部分中，研究表明对于嵌入欧拉属 k 的曲面或嵌入属 k 的不可定向曲面至关重要的图形是由 3 个相连的分量构建而成的，这些分量被称为跳板和级联。在第二部分中，将对欧拉属 2 的所有级联进行分类。因此，可以得到将图形嵌入克莱因瓶的连通性 2 的完整障碍列表。
{"title":"Excluded minors for the Klein bottle II. Cascades","authors":"Bojan Mohar , Petr Škoda","doi":"10.1016/j.jctb.2023.12.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.12.006","url":null,"abstract":"<div><p>Graphs that are critical (minimal excluded minors) for embeddability in surfaces are studied. In Part I, it was shown that graphs that are critical for embeddings into surfaces of Euler genus <em>k</em><span> or for embeddings into nonorientable surface of genus </span><em>k</em><span><span> are built from 3-connected components, called hoppers and cascades. In Part II, all cascades for Euler genus 2 are classified. As a consequence, the complete list of obstructions of connectivity 2 for embedding graphs into the </span>Klein bottle is obtained.</span></p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-01-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139433917","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}
Pub Date : 2024-01-05DOI: 10.1016/j.jctb.2023.12.003
Oscar Defrain , Jean-Florent Raymond
Graphs of bounded degeneracy are known to contain induced paths of order when they contain a path of order n, as proved by Nešetřil and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray conjectured that this bound could be improved to for some constant depending on the degeneracy.
We disprove this conjecture by constructing, for arbitrarily large values of n, a graph that is 2-degenerate, has a path of order n, and where all induced paths have order . We also show that the graphs we construct have linearly bounded coloring numbers.
已知有界退化图在包含阶数为 n 的路径时，会包含阶数为Ω(loglogn)的诱导路径，Nešetřil 和 Ossona de Mendez（2012 年）证明了这一点。2016年，Esperet、Lemoine和Maffray猜想，对于某个常数c>0（取决于退化程度），这个约束可以改进为Ω((logn)c)。我们推翻了这个猜想，为任意大的n值构造了一个图，它是2退化的，有一条阶数为n的路径，并且所有诱导路径的阶数都是O((loglogn)2)。我们还证明了我们构建的图具有线性有界着色数。
{"title":"Sparse graphs without long induced paths","authors":"Oscar Defrain , Jean-Florent Raymond","doi":"10.1016/j.jctb.2023.12.003","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.12.003","url":null,"abstract":"<div><p>Graphs of bounded degeneracy are known to contain induced paths of order <span><math><mi>Ω</mi><mo>(</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></math></span> when they contain a path of order <em>n</em>, as proved by Nešetřil and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray conjectured that this bound could be improved to <span><math><mi>Ω</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mi>c</mi></mrow></msup><mo>)</mo></math></span> for some constant <span><math><mi>c</mi><mo>></mo><mn>0</mn></math></span> depending on the degeneracy.</p><p>We disprove this conjecture by constructing, for arbitrarily large values of <em>n</em>, a graph that is 2-degenerate, has a path of order <em>n</em>, and where all induced paths have order <span><math><mi>O</mi><mo>(</mo><msup><mrow><mo>(</mo><mi>log</mi><mo></mo><mi>log</mi><mo></mo><mi>n</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>)</mo></math></span>. We also show that the graphs we construct have linearly bounded coloring numbers.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139107742","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}
Pub Date : 2024-01-05DOI: 10.1016/j.jctb.2023.12.004
Dániel Garamvölgyi , Tibor Jordán , Csaba Király
We consider two types of matroids defined on the edge set of a graph G: count matroids , in which independence is defined by a sparsity count involving the parameters k and ℓ, and the -cofactor matroid , in which independence is defined by linear independence in the cofactor matrix of G. We show, for each pair , that if G is sufficiently highly connected, then has maximum rank for all , and the matroid is connected. These results unify and extend several previous results, including theorems of Nash-Williams and Tutte (), and Lovász and Yemini (). We also prove that if G is highly connected, then the vertical connectivity of is also high.
We use these results to generalize Whitney's celebrated result on the graphic matroid of G (which corresponds to ) to all count matroids and to the -cofactor matroid: if G is highly connected, depending on k and ℓ, then the count matroid uniquely determines G; and similarly, if G is 14-connected, then its -cofactor matroid uniquely determines G. We also derive similar results for the t-fold union of the -cofactor matroid, and use them to prove that every 24-connected graph has a spanning tree T for which is 3-connected, whi
我们考虑了两种定义在图 G 边集上的矩阵：计数矩阵 Mk,ℓ(G)，其中独立性由涉及参数 k 和 ℓ 的稀疏性计数定义；C21-协因矩阵 C(G)，其中独立性由 G 的协因矩阵中的线性独立性定义。我们证明，对于每一对 (k,ℓ)，如果 G 具有足够高的连通性，那么对于所有 e∈E(G)，G-e 都具有最大秩，并且矩阵 Mk,ℓ(G) 是连通的。这些结果统一并扩展了之前的一些结果，包括纳什-威廉姆斯和图特（k=ℓ=1）以及洛瓦兹和叶米尼（k=2,ℓ=3）的定理。我们还证明，如果 G 的连通性很高，那么 C(G) 的垂直连通性也很高。我们利用这些结果将惠特尼关于 G 的图形矩阵（对应于 M1,1(G)）的著名结果推广到所有计数矩阵和 C21 因子矩阵：如果 G 是高度连通的，则计数矩阵 Mk,ℓ(G) 唯一决定 G；同样，如果 G 是 14 连通的，则其 C21 因子矩阵 C(G) 唯一决定 G。我们还推导出了 C21 因子矩阵的 t 折叠联合的类似结果，并用它们证明了每个 24 连接图都有一棵生成树 T，而 G-E(T)是 3 连接的，这验证了克里塞尔猜想的一种情况。
{"title":"Count and cofactor matroids of highly connected graphs","authors":"Dániel Garamvölgyi , Tibor Jordán , Csaba Király","doi":"10.1016/j.jctb.2023.12.004","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.12.004","url":null,"abstract":"<div><p>We consider two types of matroids defined on the edge set of a graph <em>G</em>: count matroids <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, in which independence is defined by a sparsity count involving the parameters <em>k</em> and <em>ℓ</em>, and the <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-cofactor matroid <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, in which independence is defined by linear independence in the cofactor matrix of <em>G</em>. We show, for each pair <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>ℓ</mi><mo>)</mo></math></span>, that if <em>G</em> is sufficiently highly connected, then <span><math><mi>G</mi><mo>−</mo><mi>e</mi></math></span> has maximum rank for all <span><math><mi>e</mi><mo>∈</mo><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, and the matroid <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is connected. These results unify and extend several previous results, including theorems of Nash-Williams and Tutte (<span><math><mi>k</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span>), and Lovász and Yemini (<span><math><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mi>ℓ</mi><mo>=</mo><mn>3</mn></math></span>). We also prove that if <em>G</em> is highly connected, then the vertical connectivity of <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is also high.</p><p>We use these results to generalize Whitney's celebrated result on the graphic matroid of <em>G</em> (which corresponds to <span><math><msub><mrow><mi>M</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span>) to all count matroids and to the <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-cofactor matroid: if <em>G</em> is highly connected, depending on <em>k</em> and <em>ℓ</em>, then the count matroid <span><math><msub><mrow><mi>M</mi></mrow><mrow><mi>k</mi><mo>,</mo><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> uniquely determines <em>G</em>; and similarly, if <em>G</em> is 14-connected, then its <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-cofactor matroid <span><math><mi>C</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> uniquely determines <em>G</em>. We also derive similar results for the <em>t</em>-fold union of the <span><math><msubsup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow><mrow><mn>1</mn></mrow></msubsup></math></span>-cofactor matroid, and use them to prove that every 24-connected graph has a spanning tree <em>T</em> for which <span><math><mi>G</mi><mo>−</mo><mi>E</mi><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is 3-connected, whi","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2024-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0095895623001120/pdfft?md5=3aa4475308b3f1d90b43521f41db45ba&pid=1-s2.0-S0095895623001120-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139107741","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-15DOI: 10.1016/j.jctb.2023.12.002
Noga Alon , Péter Frankl
We determine the maximum possible number of edges of a graph with n vertices, matching number at most s and clique number at most k for all admissible values of the parameters.
对于所有允许的参数值，我们确定了具有n个顶点的图的最大可能边数，匹配数最多为s，团数最多为k。
{"title":"Turán graphs with bounded matching number","authors":"Noga Alon , Péter Frankl","doi":"10.1016/j.jctb.2023.12.002","DOIUrl":"10.1016/j.jctb.2023.12.002","url":null,"abstract":"<div><p><span>We determine the maximum possible number of edges of a graph with </span><em>n</em><span> vertices, matching number at most </span><em>s</em> and clique number at most <em>k</em> for all admissible values of the parameters.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138634766","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}
Pub Date : 2023-12-11DOI: 10.1016/j.jctb.2023.11.004
Tung Nguyen , Alex Scott , Paul Seymour
Two subgraphs 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 there exists such that if G has chromatic number at least d, and does not contain the complete graph as a subgraph, then there are anticomplete subgraphs , 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 , is not enough to guarantee two anticomplete subgraphs both with large minimum degree; but it works if instead of excluding we exclude the complete bipartite graph . More exactly: for all there exists such that if G has minimum degree at least d, and does not contain the complete bipartite graph as a subgraph, then there are two anticomplete subgraphs both with minimum degree at least c.
如果图 G 的两个子图 A,B 的顶点不相交，并且没有连接它们的边，那么这两个子图就是反完全子图。如果 G 是一个具有有界簇数和足够大色度数的图，那么它是否真的有两个都具有大色度数的反完全子图？这是 El-Zahar 和 Erdős 在 1986 年提出的问题，至今仍未解决。如果是这样，那么至少应该存在两个最小度数都很大的反完全子图，这就是我们的结果之一。首先是强化：我们可以要求两个子图中的一个具有大色度数：即对于所有 t,c≥1，存在 d≥1，使得如果 G 的色度数至少为 d，并且不包含完整图 Kt 作为子图，那么存在反完整子图 A,B，其中 A 的最小度数至少为 c，B 的色度数至少为 c。其次，我们来看看如果用 G 具有足够大的最小度这一假设来代替 G 具有足够大的色度数这一假设，会出现什么情况。这一点，加上排除 Kt，还不足以保证两个反完全子图都具有很大的最小度；但是如果我们不排除 Kt，而是排除完整的双向图 Kt,t，就能做到这一点。更确切地说：对于所有 t,c≥1，存在 d≥1，使得如果 G 的最小度至少为 d，并且不包含完整双方形图 Kt,t 作为子图，那么存在两个最小度至少为 c 的反完全子图。
{"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><p>Two subgraphs <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span> of a graph <em>G</em> are <em>anticomplete</em> if they are vertex-disjoint and there are no edges joining them. Is it true that if <em>G</em><span> 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.</span></p><p>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 <span><math><mi>t</mi><mo>,</mo><mi>c</mi><mo>≥</mo><mn>1</mn></math></span> there exists <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span> such that if <em>G</em> has chromatic number at least <em>d</em>, and does not contain the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> as a subgraph, then there are anticomplete subgraphs <span><math><mi>A</mi><mo>,</mo><mi>B</mi></math></span>, where <em>A</em> has minimum degree at least <em>c</em> and <em>B</em> has chromatic number at least <em>c</em>.</p><p>Second, we look at what happens if we replace the hypothesis that <em>G</em> has sufficiently large chromatic number with the hypothesis that <em>G</em> has sufficiently large minimum degree. This, together with excluding <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, is <em>not</em> enough to guarantee two anticomplete subgraphs both with large minimum degree; but it works if instead of excluding <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span><span> we exclude the complete bipartite graph </span><span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>. More exactly: for all <span><math><mi>t</mi><mo>,</mo><mi>c</mi><mo>≥</mo><mn>1</mn></math></span> there exists <span><math><mi>d</mi><mo>≥</mo><mn>1</mn></math></span> such that if <em>G</em> has minimum degree at least <em>d</em>, and does not contain the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span> as a subgraph, then there are two anticomplete subgraphs both with minimum degree at least <em>c</em>.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"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}
Pub Date : 2023-12-05DOI: 10.1016/j.jctb.2023.11.006
Yan Wang , Hehui Wu
In this paper, we prove that every graph with average degree at least has a vertex partition into two parts, such that one part has average degree at least s, and the other part has average degree at least t. This solves a conjecture of Csóka, Lo, Norin, Wu and Yepremyan.
{"title":"Graph partitions under average degree constraint","authors":"Yan Wang , Hehui Wu","doi":"10.1016/j.jctb.2023.11.006","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.11.006","url":null,"abstract":"<div><p>In this paper, we prove that every graph with average degree at least <span><math><mi>s</mi><mo>+</mo><mi>t</mi><mo>+</mo><mn>2</mn></math></span> has a vertex partition into two parts, such that one part has average degree at least <em>s</em>, and the other part has average degree at least <em>t</em>. This solves a conjecture of Csóka, Lo, Norin, Wu and Yepremyan.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138484338","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}
Pub Date : 2023-11-29DOI: 10.1016/j.jctb.2023.11.005
Sepehr Hajebi , Yanjia Li , Sophie Spirkl
We prove that every -free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where denotes the t-vertex path, and for graphs , we say G is H-free if no induced subgraph of G is isomorphic to H).
More generally, let us say a class of graphs is η-bounded if there exists a function such that for every graph , where denotes smallest cardinality of a hitting set of all maximum stable sets in G, and is the clique number of G. Also, 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 can be for a 3-colourable graph G), and motivated by a number of meaningful similarities to χ-boundedness, namely,
•
given a graph G, we have for every induced subgraph H of G if and only if G is perfect;
•
there are graphs G with both and the girth of G arbitrarily large; and
•
if is a hereditary class of graphs which is polynomially η-bounded, then satisfies the Erdős-Hajnal conjecture.
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.
{"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><p>We prove that every <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span><span>-free graph of bounded clique number contains a small hitting set of all its maximum stable sets (where </span><span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span> denotes the <em>t</em>-vertex path, and for graphs <span><math><mi>G</mi><mo>,</mo><mi>H</mi></math></span>, we say <em>G</em> is <em>H-free</em><span> if no induced subgraph of </span><em>G</em> is isomorphic to <em>H</em>).</p><p>More generally, let us say a class <span><math><mi>C</mi></math></span> of graphs is <em>η-bounded</em> if there exists a function <span><math><mi>h</mi><mo>:</mo><mi>N</mi><mo>→</mo><mi>N</mi></math></span> such that <span><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></span> for every graph <span><math><mi>G</mi><mo>∈</mo><mi>C</mi></math></span>, where <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> denotes smallest cardinality of a hitting set of all maximum stable sets in <em>G</em>, and <span><math><mi>ω</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the clique number of <em>G</em>. Also, <span><math><mi>C</mi></math></span> is said to be <em>polynomially η-bounded</em> if in addition <em>h</em> can be chosen to be a polynomial.</p><p>We introduce <em>η</em>-boundedness inspired by a question of Alon (asking how large <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> can be for a 3-colourable graph <em>G</em>), and motivated by a number of meaningful similarities to <em>χ</em>-boundedness, namely,</p><ul><li><span>•</span><span><p>given a graph <em>G</em>, we have <span><math><mi>η</mi><mo>(</mo><mi>H</mi><mo>)</mo><mo>≤</mo><mi>ω</mi><mo>(</mo><mi>H</mi><mo>)</mo></math></span> for every induced subgraph <em>H</em> of <em>G</em> if and only if <em>G</em> is perfect;</p></span></li><li><span>•</span><span><p>there are graphs <em>G</em> with both <span><math><mi>η</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and the girth of <em>G</em> arbitrarily large; and</p></span></li><li><span>•</span><span><p>if <span><math><mi>C</mi></math></span> is a hereditary class of graphs which is polynomially <em>η</em>-bounded, then <span><math><mi>C</mi></math></span> satisfies the Erdős-Hajnal conjecture.</p></span></li></ul> The second bullet above in particular suggests an analogue of the Gyárfás-Sumner conjecture, that the class of all <em>H</em>-free graphs is <em>η</em>-bounded if (and only if) <em>H</em> is a forest. Like <em>χ</em>-boundedness, the case where <em>H</em> is a star is easy to verify, and we prove two non-trivial extensions of this: <em>H</em>-free graphs are <em>η</em>-bounded if (1) <em>H</em> has a vertex incident with all edges of <em>H</em>, or (2) <em>H</em> can be obtained from a star by subdividing at most one edge, exactly once.<p>Unlike <em>χ</em>-boundedness","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"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}
Pub Date : 2023-11-29DOI: 10.1016/j.jctb.2023.10.009
Jakub Kozik , Piotr Micek , William T. Trotter
We show that height h posets that have planar cover graphs have dimension . Previously, the best upper bound was . Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes as a minor.
{"title":"Dimension is polynomial in height for posets with planar cover graphs","authors":"Jakub Kozik , Piotr Micek , William T. Trotter","doi":"10.1016/j.jctb.2023.10.009","DOIUrl":"10.1016/j.jctb.2023.10.009","url":null,"abstract":"<div><p>We show that height <em>h</em><span> posets that have planar cover graphs have dimension </span><span><math><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>6</mn></mrow></msup><mo>)</mo></math></span>. Previously, the best upper bound was <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>O</mi><mo>(</mo><msup><mrow><mi>h</mi></mrow><mrow><mn>3</mn></mrow></msup><mo>)</mo></mrow></msup></math></span><span>. Planarity plays a key role in our arguments, since there are posets such that (1) dimension is exponential in height and (2) the cover graph excludes </span><span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> as a minor.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138455878","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}
We prove that for fixed k, every k-uniform hypergraph on n vertices and of minimum codegree at least contains every spanning tight k-tree of bounded vertex degree as a subgraph. This generalises a well-known result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions.
{"title":"Dirac-type conditions for spanning bounded-degree hypertrees","authors":"Matías Pavez-Signé , Nicolás Sanhueza-Matamala , Maya Stein","doi":"10.1016/j.jctb.2023.11.002","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.11.002","url":null,"abstract":"<div><p>We prove that for fixed <em>k</em>, every <em>k</em><span>-uniform hypergraph on </span><em>n</em> vertices and of minimum codegree at least <span><math><mi>n</mi><mo>/</mo><mn>2</mn><mo>+</mo><mi>o</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> contains every spanning tight <em>k</em>-tree of bounded vertex degree as a subgraph. This generalises a well-known result of Komlós, Sárközy and Szemerédi for graphs. Our result is asymptotically sharp. We also prove an extension of our result to hypergraphs that satisfy some weak quasirandomness conditions.</p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138430649","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}
Pub Date : 2023-11-22DOI: 10.1016/j.jctb.2023.10.010
Marthe Bonamy , Michelle Delcourt , Richard Lang , Luke Postle
The famous List Colouring Conjecture from the 1970s states that for every graph G the chromatic index of G is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph G with sufficiently large maximum degree Δ and minimum degree , the following holds: for every assignment L of lists of colours to the edges of G, such that for each edge , there is an L-edge-colouring of G. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, k-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.
{"title":"Edge-colouring graphs with local list sizes","authors":"Marthe Bonamy , Michelle Delcourt , Richard Lang , Luke Postle","doi":"10.1016/j.jctb.2023.10.010","DOIUrl":"https://doi.org/10.1016/j.jctb.2023.10.010","url":null,"abstract":"<div><p>The famous List Colouring Conjecture from the 1970s states that for every graph <em>G</em> the chromatic index of <em>G</em><span> is equal to its list chromatic index. In 1996 in a seminal paper, Kahn proved that the List Colouring Conjecture holds asymptotically. Our main result is a local generalization of Kahn's theorem. More precisely, we show that, for a graph </span><em>G</em><span> with sufficiently large maximum degree Δ and minimum degree </span><span><math><mi>δ</mi><mo>≥</mo><msup><mrow><mi>ln</mi></mrow><mrow><mn>25</mn></mrow></msup><mo></mo><mi>Δ</mi></math></span>, the following holds: for every assignment <em>L</em> of lists of colours to the edges of <em>G</em>, such that <span><math><mo>|</mo><mi>L</mi><mo>(</mo><mi>e</mi><mo>)</mo><mo>|</mo><mo>≥</mo><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo><mo>⋅</mo><mi>max</mi><mo></mo><mrow><mo>{</mo><mi>deg</mi><mo></mo><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mi>deg</mi><mo></mo><mo>(</mo><mi>v</mi><mo>)</mo><mo>}</mo></mrow></math></span> for each edge <span><math><mi>e</mi><mo>=</mo><mi>u</mi><mi>v</mi></math></span>, there is an <em>L</em>-edge-colouring of <em>G</em>. Furthermore, Kahn showed that the List Colouring Conjecture holds asymptotically for linear, <em>k</em><span>-uniform hypergraphs, and recently Molloy generalized Kahn's original result to correspondence colouring as well as its hypergraph generalization. We prove local versions of all of these generalizations by showing a weighted version that simultaneously implies all of our results.</span></p></div>","PeriodicalId":54865,"journal":{"name":"Journal of Combinatorial Theory Series B","volume":null,"pages":null},"PeriodicalIF":1.4,"publicationDate":"2023-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138430648","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}
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