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A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs 部分立方体的立方多项式与其交叉图的小块多项式之间的关系
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-04-23 DOI: 10.1002/jgt.23099
Yan-Ting Xie, Yong-De Feng, Shou-Jun Xu

Partial cubes are the graphs which can be embedded into hypercubes. The cube polynomial of a graph � � G is a counting polynomial of induced hypercubes of � � G, which is defined as � � C� � (� � G� � ,� � x� � )� � � � � � i� � � � 0� � α� � i� � (� � G� � )� � x� � i, where � � α� � i� � (� � G� � ) is the number of induced � � i-cubes (hypercubes of dimension � � i) of � � G. The clique polynomial of � � G is defined as � � C� � l� � (� � G� � ,� � x� �

局部立方体是指可以嵌入超立方体的图形。图的立方多项式是Ⅳ的诱导超立方的计数多项式,定义为Ⅳ,其中Ⅳ是Ⅳ的诱导-立方(维数为Ⅳ的超立方)的个数。 Ⅳ的簇多项式定义为Ⅳ,其中()是Ⅳ中簇的个数。等价地,恰好是 的补集的独立性多项式。 部分立方体的交叉图是其顶点对应于 的-类的图,并且当且仅当两个-类在 中交叉时,它们在 中相邻。 在本文中,我们证明对于部分立方体 ,并且当且仅当 是一个中值图时,等价成立。由于每个图都可以表示为中值图的交叉图,上述必要且充分的结果表明,对中值图的立方多项式的研究可以转化为对一般图的簇多项式的研究(等同于对其补集的独立性多项式的研究)。此外,我们还推翻了中值图的立方多项式是单模态的猜想。
{"title":"A relation between the cube polynomials of partial cubes and the clique polynomials of their crossing graphs","authors":"Yan-Ting Xie,&nbsp;Yong-De Feng,&nbsp;Shou-Jun Xu","doi":"10.1002/jgt.23099","DOIUrl":"10.1002/jgt.23099","url":null,"abstract":"<p>Partial cubes are the graphs which can be embedded into hypercubes. The <i>cube polynomial</i> of a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a counting polynomial of induced hypercubes of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, which is defined as <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>x</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>≔</mo>\u0000 \u0000 <msub>\u0000 <mo>∑</mo>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 \u0000 <mo>⩾</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow>\u0000 </msub>\u0000 \u0000 <msub>\u0000 <mi>α</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <msup>\u0000 <mi>x</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msup>\u0000 </mrow></math>, where <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>α</mi>\u0000 \u0000 <mi>i</mi>\u0000 </msub>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is the number of induced <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 </mrow></math>-cubes (hypercubes of dimension <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>i</mi>\u0000 </mrow></math>) of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>. The <i>clique polynomial</i> of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is defined as <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>C</mi>\u0000 \u0000 <mi>l</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>x</mi>\u0000 </mrow>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140803919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Flexible list colorings: Maximizing the number of requests satisfied 灵活的列表着色:最大限度地满足请求数量
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-04-18 DOI: 10.1002/jgt.23103
Hemanshu Kaul, Rogers Mathew, Jeffrey A. Mudrock, Michael J. Pelsmajer

Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose � � 0� � � � ϵ� � � � 1, � � G is a graph, � � L is a list assignment for � � G, and � � r is a function with nonempty domain � � D� � � � V� � (� � G� � ) such that � � r� � (� � v� � )� � � � L� � (� � v� � ) for each � � v� � � � D (� � r is called a request of � � L). The triple � � (� � G� � ,� � L� � ,� � r� � ) is � � ϵ-satisfiable if there exists a proper � � L

灵活列表着色由 Dvořák、Norin 和 Postle 于 2019 年提出。假设 ,是一个图,是对 ,的列表赋值,并且是一个具有非空域的函数,这样对于每个 ( 称为 )的请求。如果存在一个适当的 ,且至少对...中的顶点而言,...是可满足的,那么这个三元组就是...可满足的。德沃夏克(Dvořák)等人曾证明,如果是质数,是退化图,并且是域大小为 1 的请求,那么只要是分配,就是可满足的。在本文中,我们将这一结果扩展到所有双向-退化图。关于灵活列表着色的文献往往侧重于证明对于一个固定的图,存在一个这样的-灵活,但很自然的是,我们试图找到最大可能的-灵活。为此,我们改进了德沃夏克等人的一项成果,证明了-退化图是-灵活的。在追求图的最大-柔性时,我们观察到,当且仅当 ,为 ,的霍尔比时,图对于任何都不是-柔性的,因此我们开始研究图的列表柔性数,它是-柔性的最小值。我们研究了图的列表柔性数、列表色度数、列表包装数和退化性之间的关系和联系。
{"title":"Flexible list colorings: Maximizing the number of requests satisfied","authors":"Hemanshu Kaul,&nbsp;Rogers Mathew,&nbsp;Jeffrey A. Mudrock,&nbsp;Michael J. Pelsmajer","doi":"10.1002/jgt.23103","DOIUrl":"10.1002/jgt.23103","url":null,"abstract":"<p>Flexible list coloring was introduced by Dvořák, Norin, and Postle in 2019. Suppose <span></span><math>\u0000 \u0000 <mrow>\u0000 <mn>0</mn>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mi>ϵ</mi>\u0000 \u0000 <mo>≤</mo>\u0000 \u0000 <mn>1</mn>\u0000 </mrow></math>, <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a graph, <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow></math> is a list assignment for <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, and <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow></math> is a function with nonempty domain <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>⊆</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> such that <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>L</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>v</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> for each <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>v</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow></math> (<span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow></math> is called a request of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 </mrow></math>). The triple <span></span><math>\u0000 \u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>L</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>r</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> is <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>ϵ</mi>\u0000 </mrow></math>-satisfiable if there exists a proper <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>L</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140627463","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Graphs with girth 9 and without longer odd holes are 3-colourable 周长为 9 且没有较长奇数孔的图形是 3 可取的
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-04-11 DOI: 10.1002/jgt.23101
Yan Wang, Rong Wu

For a number � � l� � � � 2, let � � G� � l denote the family of graphs which have girth � � 2� � l� � +� � 1 and have no odd hole with length greater than � � 2� � l� � +� � 1. Wu, Xu and Xu conjectured that every graph in � � � � l� � � � 2� � G� � l is 3-colourable. Chudnovsky et al., Wu et al., and Chen showed that every graph in � � G� � 2, � � G� � 3 and � � � � l� � � � 5� � G� � l is 3-colourable, respectively. In this paper, we prove that every graph in � � G� � 4 is 3-colourable. This confirms Wu, Xu and Xu's conjecture.

对于一个数 ,让 表示有周长且没有长度大于 的奇数洞的图族。 吴、徐和徐猜想,在 的每一个图都是 3 可容的。Chudnovsky 等人、Wu 等人和 Chen 分别证明了 、 和 中的每个图都是 3 可容的。在本文中,我们证明了每个 in 的图都是 3 有利的。这证实了 Wu、Xu 和 Xu 的猜想。
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引用次数: 0
Best possible upper bounds on the restrained domination number of cubic graphs 立方图的约束支配数的最佳上限
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-04-11 DOI: 10.1002/jgt.23095
Boštjan Brešar, Michael A. Henning

A dominating set in a graph � � G is a set � � S of vertices such that every vertex in � � V� � (� � G� � )� � � � S is adjacent to a vertex in � � S. A restrained dominating set of � � G is a dominating set � � S with the additional restraint that the graph � � G� � � � S obtained by removing all vertices in � � S is isolate-free. The domination number � � γ� � (� � G� � ) and the restrained domination number � � γ� � r� � (� � G� � ) are the minimum cardinalities of a dominating set and restrained dominating set, respectively, of � � G. Let � � G be a cubic graph of order � � n. A classical result of Reed states that � �

图中的支配集是这样一个顶点集合:图中的每个顶点都与图中的一个顶点相邻。限制支配集是一个支配集,其附加限制条件是移除所有 in 中的顶点后得到的图是无孤立的。支配数和受约束支配数分别是支配集和受约束支配集的最小心数。 假设是一个阶为 的立方图。里德的一个经典结果表明, ,而这个界限是可能的最佳界限。要确定......的受约束支配数的最佳可能上限则更具挑战性,我们将证明......。
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引用次数: 0
Ramsey numbers for multiple copies of sparse graphs 稀疏图形多副本的拉姆齐数
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-04-11 DOI: 10.1002/jgt.23100
Aurelio Sulser, Miloš Trujić

For a graph � � H and an integer � � n, we let � � n� � H denote the disjoint union of � � n copies of � � H. In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for � � n� � H, one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant � � c� � =� � c� � (� � H� � ) such that � � r� � (� � n� � H� � )� � =� � (� � 2� � � � H� � � � � � α� � (� � H� � )� � )� � n� � +� � c, provided

对于一个图和一个整数 ,我们用 表示其副本的不相交联合。 1975 年,伯尔、厄多斯和斯宾塞开始研究拉姆齐数,拉姆齐数是目前已知拉姆齐数的少数实例之一。他们证明,只要拉姆齐数足够大,就会有一个常数使得 , 。随后,伯尔给出了一种隐含的计算方法,并指出这种长期行为发生在......的三倍指数时。最近,布契奇和苏达科夫重新提出了这个问题,并建立了一个基本严密的约束,表明当副本数仅为单指数时,这种行为已经出现。在稀疏图的情况下,我们提供了明显更强的约束,最明显的是有界最大度。这些约束与当前最先进的约束是相关的,而且(在某种程度上)是紧密的。我们的方法依赖于 Graham、Rödl 和 Ruciński 的一个漂亮的经典证明,重点是为有界度图开发一种高效的吸收方法。
{"title":"Ramsey numbers for multiple copies of sparse graphs","authors":"Aurelio Sulser,&nbsp;Miloš Trujić","doi":"10.1002/jgt.23100","DOIUrl":"10.1002/jgt.23100","url":null,"abstract":"<p>For a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math> and an integer <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>, we let <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>H</mi>\u0000 </mrow></math> denote the disjoint union of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math> copies of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>H</mi>\u0000 </mrow></math>. In 1975, Burr, Erdős and Spencer initiated the study of Ramsey numbers for <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>H</mi>\u0000 </mrow></math>, one of few instances for which Ramsey numbers are now known precisely. They showed that there is a constant <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>c</mi>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mi>c</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> such that <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>r</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 \u0000 <mi>H</mi>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>=</mo>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mrow>\u0000 <mn>2</mn>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>∣</mo>\u0000 \u0000 <mo>−</mo>\u0000 \u0000 <mi>α</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>H</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mi>n</mi>\u0000 \u0000 <mo>+</mo>\u0000 \u0000 <mi>c</mi>\u0000 </mrow></math>, provided <span></span><math>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23100","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572898","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bisimplicial separators 二等分隔符
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-04-11 DOI: 10.1002/jgt.23098
Martin Milanič, Irena Penev, Nevena Pivač, Kristina Vušković

A minimal separator of a graph � � G is a set � � S� � � � V� � (� � G� � ) such that there exist vertices � � a� � ,� � b� � � � V� � (� � G� � )� � � � S with the property that � � S separates � � a from � � b in � � G, but no proper subset of � � S does. For an integer � � k� � � � 0, we say that a minimal separator is � � k-simplicial if it can be covered by � � k cliques and denote by � � G� � k the class of all graphs in which each minimal separator is � � k-simplicial. We show that for each � � k� �

一个图的最小分隔符是这样一个集合,即存在这样的顶点,它们具有从 、 中分隔开来的属性,但没有适当的子集这样做。对于整数 ,如果最小分隔符可以被小块覆盖,我们就说它是-简单的,并用所有每个最小分隔符都是-简单的图的类来表示。我们证明,对于每个 ,该类在诱导最小分隔符下是封闭的,并以此证明最大权重稳定集问题可以在多项式时间内求解.我们还给出了......的最小禁止诱导最小数的完整列表。接下来,我们证明,对于......中的每个非空图,都有一个简单顶点,也就是说,有一个顶点的邻域是小群的联合;我们推导出,对于......中的图,最大权重小群问题可以在多项式时间内求解。此外,我们还证明了,对于......,识别......中的图是 NP 难的;识别......中的图的时间复杂度尚不清楚。我们还证明,对于......中的图,最大克立(Maximum Clique)问题是 NP-hard。最后,我们证明了 in 中无菱形图的分解定理(其中菱形图是通过删除一条边得到的图),并利用该定理得到了针对 , 中无菱形图的顶点着色问题和识别问题的多项式时间算法,以及针对该类图的最大权重簇问题和最大权重稳定集问题的改进运行时间。
{"title":"Bisimplicial separators","authors":"Martin Milanič,&nbsp;Irena Penev,&nbsp;Nevena Pivač,&nbsp;Kristina Vušković","doi":"10.1002/jgt.23098","DOIUrl":"10.1002/jgt.23098","url":null,"abstract":"<p>A <i>minimal separator</i> of a graph <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math> is a set <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 \u0000 <mo>⊆</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow></math> such that there exist vertices <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 \u0000 <mo>,</mo>\u0000 \u0000 <mi>b</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>V</mi>\u0000 \u0000 <mrow>\u0000 <mo>(</mo>\u0000 \u0000 <mi>G</mi>\u0000 \u0000 <mo>)</mo>\u0000 </mrow>\u0000 \u0000 <mo>⧹</mo>\u0000 \u0000 <mi>S</mi>\u0000 </mrow></math> with the property that <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> separates <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow></math> from <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>b</mi>\u0000 </mrow></math> in <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow></math>, but no proper subset of <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>S</mi>\u0000 </mrow></math> does. For an integer <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 <mo>≥</mo>\u0000 \u0000 <mn>0</mn>\u0000 </mrow></math>, we say that a minimal separator is <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-<i>simplicial</i> if it can be covered by <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math> cliques and denote by <span></span><math>\u0000 \u0000 <mrow>\u0000 <msub>\u0000 <mi>G</mi>\u0000 \u0000 <mi>k</mi>\u0000 </msub>\u0000 </mrow></math> the class of all graphs in which each minimal separator is <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-simplicial. We show that for each <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 \u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23098","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140572914","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Edge-minimum saturated k -planar drawings 边缘最小饱和 k 平面绘图
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-03-28 DOI: 10.1002/jgt.23097
Steven Chaplick, Fabian Klute, Irene Parada, Jonathan Rollin, Torsten Ueckerdt

For a class � � D of drawings of loopless (multi-)graphs in the plane, a drawing � � D� � � � D is saturated when the addition of any edge to � � D results in � � D� � � � � � D—this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on � � k-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most � � k times, and the classes � � D of all � � k-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated � � k-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all � � n-vertex saturated � � k-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest � � n-vertex saturated � � k-planar drawings have � � 2� � k

对于平面中无循环(多)图的一类 D${mathscr{D}}$ 图、当在 D$D$ 中添加任何边都会导致 D′∉D${D}^{^{prime} }notin {mathscr{D}}$ 时,图 D∈D$Din {mathscr{D}}$ 就是饱和图--这类似于图兰和厄多斯、哈伊纳尔和穆恩提出的图类中的饱和图。我们关注的是 k$k$-planar 绘图,即在平面上绘制的、每条边最多交叉 k$k$ 次的图形,以及所有 k$k$-planar 绘图的类 D${mathscr{D}}$,这些类遵守一系列限制条件,例如没有交叉的附带边、没有交叉超过一次的边对或没有交叉本身的边。虽然饱和 k$k$-planar 绘图是之前几项研究的重点,但对这些绘图的稀疏程度的严格限制却不甚了解。我们建立了一个通用框架,以确定许多自然类中所有 n$n$ 顶点饱和 k$k$ 平面图形的最小边数。例如,当入射交叉、多交叉和自交叉都被允许时,最稀疏的 n$n$-顶点饱和 k$k$-平面图有 2k-(kmod2)(n-1)$frac{2}{k-(k、对于任意 k≥4$kge 4$,最稀疏的平面图有 2(k+1)k(k-1)(n-1)$frac{2(k+1)}{k(k-1)}(n-1)$边。
{"title":"Edge-minimum saturated \u0000 \u0000 k\u0000 -planar drawings","authors":"Steven Chaplick,&nbsp;Fabian Klute,&nbsp;Irene Parada,&nbsp;Jonathan Rollin,&nbsp;Torsten Ueckerdt","doi":"10.1002/jgt.23097","DOIUrl":"10.1002/jgt.23097","url":null,"abstract":"<p>For a class <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> of drawings of loopless (multi-)graphs in the plane, a drawing <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 \u0000 <mo>∈</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow></math> is <i>saturated</i> when the addition of any edge to <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> results in <span></span><math>\u0000 \u0000 <mrow>\u0000 <msup>\u0000 <mi>D</mi>\u0000 \u0000 <mo>′</mo>\u0000 </msup>\u0000 \u0000 <mo>∉</mo>\u0000 \u0000 <mi>D</mi>\u0000 </mrow></math>—this is analogous to saturated graphs in a graph class as introduced by Turán and Erdős, Hajnal, and Moon. We focus on <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings, that is, graphs drawn in the plane where each edge is crossed at most <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math> times, and the classes <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow></math> of all <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings obeying a number of restrictions, such as having no crossing incident edges, no pair of edges crossing more than once, or no edge crossing itself. While saturated <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings are the focus of several prior works, tight bounds on how sparse these can be are not well understood. We establish a generic framework to determine the minimum number of edges among all <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>-vertex saturated <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings in many natural classes. For example, when incident crossings, multicrossings and selfcrossings are all allowed, the sparsest <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>n</mi>\u0000 </mrow></math>-vertex saturated <span></span><math>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 </mrow></math>-planar drawings have <span></span><math>\u0000 \u0000 <mrow>\u0000 <mfrac>\u0000 <mn>2</mn>\u0000 \u0000 <mrow>\u0000 <mi>k</mi>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23097","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140323923","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
The chromatic number of heptagraphs 七段体的色度数
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-03-19 DOI: 10.1002/jgt.23094
Di Wu, Baogang Xu, Yian Xu

A pentagraph is a graph without cycles of length 3 or 4 and without induced cycles of odd length at least 7, and a heptagraph is one without cycles of length less than 7 and without induced cycles of odd length at least 9. Chudnovsky and Seymour proved that every pentagraph is 3-colorable. In this paper, we show that every heptagraph is 3-colorable.

五段图是没有长度为 3 或 4 的循环且没有奇数长度至少为 7 的诱导循环的图形,七段图是没有长度小于 7 的循环且没有奇数长度至少为 9 的诱导循环的图形。Chudnovsky 和 Seymour 证明了每个五段都是 3 色的。在本文中,我们将证明每个七段都是 3 色的。
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引用次数: 0
Circular flows in mono-directed signed graphs 单向有符号图中的循环流动
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-03-19 DOI: 10.1002/jgt.23092
Jiaao Li, Reza Naserasr, Zhouningxin Wang, Xuding Zhu

In this paper, the concept of circular r $r$-flow in a mono-directed signed graph (G,σ) $(G,sigma )$ is introduced. That is a pair (D,f) $(D,f)$, where D $D$ is an orientation on G $G$ and f:E(G)(r,r) $f:E(G)to (-r,r)$ satisfies that f(e)[1,r1

本文引入了单向有符号图(G,σ)$(G,sigma )$中循环 r$r$ 流的概念。即一对 (D,f)$(D,f)$,其中 D$D$ 是 G$G$ 上的方向,f:E(G)→(-r,r)$f:E(G)to(-r,r)$满足∣f(e)∣∈[1,r-1]$| f(e)| in [1,r-1]$ for each positive edge e$e$ and ∣f(e)∣∈[0,r2-1]∪[r2+1、r)$|f(e)|in[0,frac{r}{2}-1]cup [frac{r}{2}+1,r)$对于每条负边 e$e$,总流入等于每个顶点的总流出。这就是有符号图的循环着色的对偶概念,有别于文献中研究的与有符号图相关的双向图中的循环流概念。我们首先探讨了有符号图中循环 2kk-1$frac{2k}{k-1}$ 流与 modulo k$k$-orientation 之间的联系。有符号图的循环流指数是允许循环 r$r$ 流的最小值 r$r$。我们证明了每一个 3 边连接的有符号图都允许循环 6 流,每一个 4 边连接的有符号图都允许循环 4 流。更一般地说,对于 k≥2$kge 2$,我们证明每一个 (3k-1)$(3k-1)$ 边连接的有符号图都有一个环形 2kk-1$frac{2k}{k-1}$ 流,每一个 3k$3k$ 边连接的有符号图都有一个环形 r$r$ 流,r<;2kk-1$rlt frac{2k}{k-1}$,而每个 (3k+1)$(3k+1)$ 边连接的有符号图都有一个循环的 4k+22k-1$frac{4k+2}{2k-1}$ 流。此外,(6k-2)$(6k-2)$-边连接条件被证明足以让有符号欧拉图接纳循环 4k2k-1$/frac{4k}{2k-1}$-流,将这一结果应用于平面图,我们得出结论:每个负周长至少为 6k-2$6k-2$ 的有符号双方平面图都接纳与负偶数循环 C-2k${C}_{-2k}$ 的同构。
{"title":"Circular flows in mono-directed signed graphs","authors":"Jiaao Li,&nbsp;Reza Naserasr,&nbsp;Zhouningxin Wang,&nbsp;Xuding Zhu","doi":"10.1002/jgt.23092","DOIUrl":"10.1002/jgt.23092","url":null,"abstract":"<p>In this paper, the concept of circular <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <annotation> $r$</annotation>\u0000 </semantics></math>-flow in a mono-directed signed graph <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 <mo>,</mo>\u0000 <mi>σ</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(G,sigma )$</annotation>\u0000 </semantics></math> is introduced. That is a pair <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 <mo>,</mo>\u0000 <mi>f</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $(D,f)$</annotation>\u0000 </semantics></math>, where <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>D</mi>\u0000 </mrow>\u0000 <annotation> $D$</annotation>\u0000 </semantics></math> is an orientation on <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>G</mi>\u0000 </mrow>\u0000 <annotation> $G$</annotation>\u0000 </semantics></math> and <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>f</mi>\u0000 <mo>:</mo>\u0000 <mi>E</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>G</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>→</mo>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mrow>\u0000 <mo>−</mo>\u0000 <mi>r</mi>\u0000 <mo>,</mo>\u0000 <mi>r</mi>\u0000 </mrow>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation> $f:E(G)to (-r,r)$</annotation>\u0000 </semantics></math> satisfies that <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mo>∣</mo>\u0000 <mi>f</mi>\u0000 <mrow>\u0000 <mo>(</mo>\u0000 <mi>e</mi>\u0000 <mo>)</mo>\u0000 </mrow>\u0000 <mo>∣</mo>\u0000 <mo>∈</mo>\u0000 <mrow>\u0000 <mo>[</mo>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 <mo>,</mo>\u0000 <mi>r</mi>\u0000 <mo>−</mo>\u0000 <mn>1</mn>\u0000 ","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.9,"publicationDate":"2024-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140181805","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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
Ramsey numbers upon vertex deletion 顶点删除后的拉姆齐数
IF 0.9 3区 数学 Q2 Mathematics Pub Date : 2024-03-18 DOI: 10.1002/jgt.23093
Yuval Wigderson

Given a graph G $G$, its Ramsey number r(� � G� � ) $r(G)$ is the minimum N $N$ so that every two-coloring of E(� � K� � N� � ) $E({K}_{N})$ contains a monochromatic copy of G $G$. It was conjectured by Conlon, Fox, and Sudakov that if one deletes a single vertex from G $G$, the Ramsey number can change by at most a constant factor. We disprove this conjecture, exhibiting an infinite family of graphs such that deleting a single vertex from each decreases the Ramsey number by a super-constant factor. One consequence of this result is the following. There exists a family of graphs {� � G� � n� � } ${{G}_{n}}$ so that in any Ramsey coloring for G

康伦、福克斯和苏达科夫猜想,如果从一个图中删除一个顶点,拉姆齐数最多只能以一个常数因子变化。我们推翻了这一猜想,展示了一个无限图族,从每个图中删除一个顶点都会使拉姆齐数减少一个超常数因子。这一结果的一个结果如下。存在这样一个图族:在任何拉姆齐着色(即没有单色副本的顶点上的一个小群的着色)中,其中一个颜色类的密度为 .
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Journal of Graph Theory
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