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Large cliques and independent sets all over the place 到处都是大集团和独立集团
Pub Date : 2020-04-09 DOI: 10.1090/proc/15323
N. Alon, M. Buci'c, B. Sudakov
We study the following question raised by Erdős and Hajnal in the early 90's. Over all $n$-vertex graphs $G$ what is the smallest possible value of $m$ for which any $m$ vertices of $G$ contain both a clique and an independent set of size $log n$? We construct examples showing that $m$ is at most $2^{2^{(loglog n)^{1/2+o(1)}}}$ obtaining a twofold sub-polynomial improvement over the upper bound of about $sqrt{n}$ coming from the natural guess, the random graph. Our (probabilistic) construction gives rise to new examples of Ramsey graphs, which while having no very large homogenous subsets contain both cliques and independent sets of size $log n$ in any small subset of vertices. This is very far from being true in random graphs. Our proofs are based on an interplay between taking lexicographic products and using randomness.
我们研究了Erdős和Hajnal在90年代初提出的问题。在所有的$n$顶点图$G$中,$m$的最小可能值是什么?对于$G$的任何$m$顶点既包含一个团又包含一个独立的大小集合$log n$ ?我们构造的例子表明$m$最多是$2^{2^{(loglog n)^{1/2+o(1)}}}$,在大约$sqrt{n}$的上界上得到了两倍的子多项式改进,这来自于自然猜测,即随机图。我们的(概率)构造产生了Ramsey图的新示例,虽然没有非常大的同质子集,但在任何小的顶点子集中都包含团和大小为$log n$的独立集。这在随机图中是远远不成立的。我们的证明是基于采用词典产品和使用随机性之间的相互作用。
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引用次数: 0
On energy ordering of vertex-disjoint bicyclic sidigraphs 顶点不相交双环边图的能量排序
Pub Date : 2020-04-08 DOI: 10.3934/math.2020430
S. Hafeez, R. Farooq
The energy and iota energy of signed digraphs are respectively defined as the sum of absolute values of real parts and sum of absolute values of imaginary parts of its eigenvalues. Recently, Yang and Wang (2018) find the energy and iota energy ordering of digraphs in D(n) and compute the maximal energy and iota energy, where D(n) denotes the set of vertex-disjoint bicyclic digraphs of a fixed order n. In this paper, we investigate the energy ordering of signed digraphs in D(n,s) and finds the maximal energy, where D(n,s) denotes the set of vertex-disjoint bicyclic sidigraphs of a fixed order n.
有符号有向图的能量和iota能量分别定义为其特征值的实部绝对值和虚部绝对值和。最近,Yang和Wang(2018)找到了D(n)中有向图的能量和iota能量排序,并计算了最大能量和iota能量,其中D(n,s)表示固定n阶的顶点不相交的双环有向图的集合。本文研究了D(n,s)中有符号有向图的能量排序,并找到了最大能量,其中D(n,s)表示固定n阶的顶点不相交的双环边图的集合。
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引用次数: 0
Iota energy orderings of bicyclic signed digraphs 双环有符号有向图的Iota能量排序
Pub Date : 2020-04-03 DOI: 10.22108/TOC.2021.126881.1805
Xiuwen Yang, Ligong Wang
The concept of energy of a signed digraph is extended to iota energy of a signed digraph. The energy of a signed digraph $S$ is defined by $E(S)=sum_{k=1}^n|text{Re}(z_k)|$, where $text{Re}(z_k)$ is the real part of eigenvalue $z_k$ and $z_k$ is the eigenvalue of the adjacency matrix of $S$ with $n$ vertices, $k=1,2,ldots,n$. Then the iota energy of $S$ is defined by $E(S)=sum_{k=1}^n|text{Im}(z_k)|$, where $text{Im}(z_k)$ is the imaginary part of eigenvalue $z_k$. In this paper, we consider a special graph class for bicyclic signed digraphs $mathcal{S}_n$ with $n$ vertices which have two vertex-disjoint signed directed even cycles. We give two iota energy orderings of bicyclic signed digraphs, one is including two positive or two negative directed even cycles, the other is including one positive and one negative directed even cycles.
将有符号有向图能量的概念推广到有符号有向图的iota能量。有符号有向图$S$的能量由$E(S)=sum_{k=1}^n|text{Re}(z_k)|$定义,其中$text{Re}(z_k)$是特征值$z_k$的实部,$z_k$是$S$具有$n$顶点的邻接矩阵的特征值,$k=1,2, $ ldots,n$。则$S$的iota能量定义为$E(S)=sum_{k=1}^n|text{Im}(z_k)|$,其中$text{Im}(z_k)$是特征值$z_k$的虚部。本文考虑了具有n个顶点的双环有符号有向图$mathcal{S}_n$具有两个顶点不相交的有符号有向偶环的特殊图类。我们给出双环有符号有向图的两种iota能量排序,一种包含两个正的或负的有向偶环,另一种包含一个正的和一个负的有向偶环。
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引用次数: 0
Ordered set partitions, Garsia-Procesi modules, and rank varieties 有序集分区、Garsia-Procesi模块和秩变量
Pub Date : 2020-04-02 DOI: 10.1090/tran/8237
Sean T. Griffin
We introduce a family of ideals $I_{n,lambda,s}$ in $mathbb{Q}[x_1,dots,x_n]$ for $lambda$ a partition of $kleq n$ and an integer $s geq ell(lambda)$. This family contains both the Tanisaki ideals $I_lambda$ and the ideals $I_{n,k}$ of Haglund-Rhoades-Shimozono as special cases. We study the corresponding quotient rings $R_{n,lambda,s}$ as symmetric group modules. When $n=k$ and $s$ is arbitrary, we recover the Garsia-Procesi modules, and when $lambda=(1^k)$ and $s=k$, we recover the generalized coinvariant algebras of Haglund-Rhoades-Shimozono. We give a monomial basis for $R_{n,lambda,s}$, unifying the monomial bases studied by Garsia-Procesi and Haglund-Rhoades-Shimozono, and realize the $S_n$-module structure of $R_{n,lambda,s}$ in terms of an action on $(n,lambda,s)$-ordered set partitions. We also prove formulas for the Hilbert series and graded Frobenius characteristic of $R_{n,lambda,s}$. We then connect our work with Eisenbud-Saltman rank varieties using results of Weyman. As an application of our work, we give a monomial basis, Hilbert series formula, and graded Frobenius characteristic formula for the coordinate ring of the scheme-theoretic intersection of a rank variety with diagonal matrices.
我们在$mathbb{Q}[x_1,dots,x_n]$中引入一个理想族$I_{n,lambda,s}$,对于$lambda$,一个分区$kleq n$和一个整数$s geq ell(lambda)$。这个家族既包含谷崎理想$I_lambda$,也包含作为特例的哈格伦德-罗兹-下野理想$I_{n,k}$。我们研究了相应的商环$R_{n,lambda,s}$作为对称群模。当$n=k$和$s$为任意时,我们恢复了Garsia-Procesi模,当$lambda=(1^k)$和$s=k$时,我们恢复了Haglund-Rhoades-Shimozono的广义协不变代数。我们统一了Garsia-Procesi和Haglund-Rhoades-Shimozono研究的单项基,给出了$R_{n,lambda,s}$的一个单项基,并通过对$(n,lambda,s)$ -有序集分区的作用实现了$R_{n,lambda,s}$的$S_n$ -模块结构。我们还证明了$R_{n,lambda,s}$的Hilbert级数和梯度Frobenius特征的公式。然后,我们使用Weyman的结果将我们的工作与Eisenbud-Saltman秩变体联系起来。作为我们工作的一个应用,我们给出了秩变与对角矩阵的方案论交的坐标环的单项式基、Hilbert级数公式和梯度Frobenius特征公式。
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引用次数: 16
Linear Programming Bounds for Spherical (k,k)-Designs 球面(k,k)-设计的线性规划界
Pub Date : 2020-04-01 DOI: 10.7546/CRABS.2020.08.02
P. Boyvalenkov
We derive general linear programming bounds for spherical $(k,k)$-designs. This includes lower bounds for the minimum cardinality and lower and upper bounds for minimum and maximum energy, respectively. As applications we obtain a universal bound in sense of Levenshtein for the minimum possible cardinality of a $(k,k)$ design for fixed dimension and $k$ and corresponding optimality result. We also discuss examples and possibilities for attaining the universal bound.
我们导出了球面$(k,k)$-设计的一般线性规划界。这包括最小基数的下界以及最小和最大能量的下界和上界。作为应用,我们得到了固定维$(k,k)$设计和$k$的最小可能基数的Levenshtein意义上的普遍界和相应的最优性结果。我们还讨论了获得全称界的例子和可能性。
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引用次数: 2
A Model for Random Braiding in Graph Configuration Spaces 图组态空间中的随机编织模型
Pub Date : 2020-04-01 DOI: 10.1093/IMRN/RNAB008
D. A. Levin, Eric Ramos, Benjamin Young
We define and study a model of winding for non-colliding particles in finite trees. We prove that the asymptotic behavior of this statistic satisfies a central limiting theorem, analogous to similar results on winding of bounded particles in the plane. We also propose certain natural open questions and conjectures, whose confirmation would provide new insights on configuration spaces of trees.
我们定义并研究了有限树中非碰撞粒子的缠绕模型。我们证明了该统计量的渐近性质满足中心极限定理,类似于平面上有界粒子缠绕的类似结果。我们还提出了一些自然开放的问题和猜想,这些问题和猜想的证实将为树木的构型空间提供新的见解。
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引用次数: 0
The translated Whitney–Lah numbers: generalizations and q-analogues 翻译后的惠特尼-拉数:概括和q-类比
Pub Date : 2020-04-01 DOI: 10.7546/nntdm.2020.26.4.80-92
M. M. Mangontarum
In this paper, we derive formulas for the translated Whitney-Lah numbers and show that they are generalizations of already-existing identities of the classical Lah numbers. q-analogues of the said formulas are also obtained for the case of the translated q-Whitney numbers.
本文导出了平移的Whitney-Lah数的公式,并证明了它们是经典Lah数的已有恒等式的推广。对于平移后的q-Whitney数,也得到了上述公式的q-类似物。
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引用次数: 1
Obstructions for Bounded Branch-depth in Matroids 矩阵中有界分支深度的障碍物
Pub Date : 2020-03-31 DOI: 10.19086/aic.24227
J. P. Gollin, Kevin Hendrey, Dillon Mayhew, Sang-il Oum
DeVos, Kwon, and Oum introduced the concept of branch-depth of matroids as a natural analogue of tree-depth of graphs. They conjectured that a matroid of sufficiently large branch-depth contains the uniform matroid $U_{n,2n}$ or the cycle matroid of a large fan graph as a minor. We prove that matroids with sufficiently large branch-depth either contain the cycle matroid of a large fan graph as a minor or have large branch-width. As a corollary, we prove their conjecture for matroids representable over a fixed finite field and quasi-graphic matroids, where the uniform matroid is not an option.
DeVos, Kwon和Oum引入了拟阵的分支深度的概念,作为图的树深度的自然类比。他们推测一个分支深度足够大的矩阵包含均匀矩阵$U_{n,2n}$或一个大扇形图的循环矩阵作为次矩阵。我们证明了具有足够大分支深度的拟阵要么包含一个大扇形图的环拟阵作为次线,要么具有较大的分支宽度。作为推论,我们证明了在固定有限域上可表示的拟阵和不能选择一致阵的拟图拟阵的猜想。
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引用次数: 1
Asymptotic Bounds on Graphical Partitions and Partition Comparability 图分区的渐近界和分区可比性
Pub Date : 2020-03-28 DOI: 10.1093/imrn/rnaa251
S. Melczer, Marcus Michelen, Somabha Mukherjee
An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than $n^{-.003}$, answering a question of Pittel. A lower bound of $n^{-1/2}$ was proven by ErdH{o}s and Richmond, and so this demonstrates that the probability decreases polynomially. Key to our argument is an asymptotic result of Pittel characterizing the joint distribution of the first rows and columns of a uniformly random partition, combined with a characterization of graphical partitions due to ErdH{o}s and Gallai. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.
如果整数分区是一个简单图的度序列,则称为图形分区。我们证明了一个大小为$n$的均匀选择分区是图形化的概率比$n^{-更快地减小到零。他回答了皮特尔的一个问题。ErdH{o}s和Richmond证明了$n^{-1/2}$的下界,从而证明了概率是多项式递减的。我们论证的关键是Pittel刻画均匀随机分区的第一列和第一行联合分布的渐近结果,并结合ErdH{o}s和Gallai对图形分区的刻画。我们的证明还暗示了两个随机选择的分区在优势顺序上具有可比性的概率的多项式上界。
{"title":"Asymptotic Bounds on Graphical Partitions and Partition Comparability","authors":"S. Melczer, Marcus Michelen, Somabha Mukherjee","doi":"10.1093/imrn/rnaa251","DOIUrl":"https://doi.org/10.1093/imrn/rnaa251","url":null,"abstract":"An integer partition is called graphical if it is the degree sequence of a simple graph. We prove that the probability that a uniformly chosen partition of size $n$ is graphical decreases to zero faster than $n^{-.003}$, answering a question of Pittel. A lower bound of $n^{-1/2}$ was proven by ErdH{o}s and Richmond, and so this demonstrates that the probability decreases polynomially. Key to our argument is an asymptotic result of Pittel characterizing the joint distribution of the first rows and columns of a uniformly random partition, combined with a characterization of graphical partitions due to ErdH{o}s and Gallai. Our proof also implies a polynomial upper bound for the probability that two randomly chosen partitions are comparable in the dominance order.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"22 4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83495170","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Decompositions of complete symmetric directed graphs into the oriented heptagons 完全对称有向图的有向七面体分解
Pub Date : 2020-03-25 DOI: 10.3906/mat-2007-54
Uğur Odabaşı
The complete symmetric directed graph of order $v$, denoted $K_{v}^*$, is the directed graph on $v$ vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair of distinct vertices $x$ and $y$. For a given directed graph, $D$, the set of all $v$ for which $K_{v}^*$ admits a $D$-decomposition is called the spectrum of $D$. There are 10 non-isomorphic orientations of a $7$-cycle (heptagon). In this paper, we completely settled the spectrum problem for each of the oriented heptagons.
$v$阶的完全对称有向图,记为$K_{v}^*$,是$v$顶点上的有向图,它包含$(x,y)$和$(y,x)$对于每一对不同的顶点$x$和$y$。对于给定的有向图$D$,其中$K_{v}^*$允许$D$分解的所有$v$的集合称为$D$的谱。$7$-环(七边形)有10个非同构取向。在本文中,我们完全解决了每个定向七面体的频谱问题。
{"title":"Decompositions of complete symmetric directed graphs into the oriented heptagons","authors":"Uğur Odabaşı","doi":"10.3906/mat-2007-54","DOIUrl":"https://doi.org/10.3906/mat-2007-54","url":null,"abstract":"The complete symmetric directed graph of order $v$, denoted $K_{v}^*$, is the directed graph on $v$ vertices that contains both arcs $(x,y)$ and $(y,x)$ for each pair of distinct vertices $x$ and $y$. For a given directed graph, $D$, the set of all $v$ for which $K_{v}^*$ admits a $D$-decomposition is called the spectrum of $D$. There are 10 non-isomorphic orientations of a $7$-cycle (heptagon). In this paper, we completely settled the spectrum problem for each of the oriented heptagons.","PeriodicalId":8442,"journal":{"name":"arXiv: Combinatorics","volume":"73 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76109804","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
期刊
arXiv: Combinatorics
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