David Conlon, Yufei Zhao
We prove that the properties of having small discrepancy and having small second eigenvalue are equivalent in Cayley graphs, extending a result of Kohayakawa, Rödl, and Schacht, who treated the abelian case. The proof relies on Grothendieck's inequality. As a corollary, we also prove that a similar result holds in all vertex-transitive graphs.
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Cayley graphs of quasirandom groups
研究拟随机群中扩张性质与简单商群的关系，通过移除Bourgain-Gamburd扩张机假设中的乘积定理，并利用冠理论证明少量随机元素高概率生成扩张Cayley图，以及拟随机群Cayley图直径的多对数上界。
Quasirandom additive sets and Cayley hypergraphs
本研究揭示加性集的Gowers一致性与超图的偏差型拟随机性的强关联，证明加性集及由其构造的Cayley型超图的多种拟随机性质在Chung-Graham-Wilson意义下多项式界等价。
We consider quasirandom properties for Cayley graphs of finite abelian groups. We show that having uniform edge-distribution (i.e., small discrepancy) and having large eigenvalue gap are equivalent properties for such Cayley graphs, even if they are sparse. This positively answers a question of Chung and Graham ["Sparse quasi-random graphs", Combinatorica 22 (2002), no. 2, 217-244] for the particular case of Cayley graphs of abelian groups, while in general the answer is negative.
We prove that any quasirandom graph with $n$ vertices and $rn$ edges can be decomposed into $n$ copies of any fixed tree with $r$ edges. The case of decomposing a complete graph establishes a conjecture of Ringel from 1963.
We show that for any fixed $α>0$, cherry-quasirandom 3-graphs of positive density and sufficiently large order $n$ with minimum vertex degree $α\binom n2$ have a tight Hamilton cycle. This solves a conjecture of Aigner-Horev and Levy.
We consider special multiclass spectral, discrepancy, degree, and codegree properties of expanding graph sequences. As we can prove equivalences and implications between them and the definition of the generalized quasirandomness of Lovász--Sós (2008), they can be regarded as generalized quasirandom properties akin to the equivalent quasirandom properties of the seminal Chung--Graham--Wilson paper (1989) in the one-class scenario. Since these properties are valid for certain deterministic graph sequences, irrespective of stochastic models, the partial implications also justify for law-dimension…
Random walks on quasirandom graphs
设G是一个n阶拟随机图，W是G上长度为α n ^2的随机游动。W所遍历的边的集合一定是拟随机图吗？这个问题是由Böttcher，Hladklund，Piguet和Taraz提出的。本文的目的就是要对这个问题给出一个肯定的回答。我们也证明了一个类似的结果随机嵌入的树。
The pantograph differential equation and its solution, the deformed exponential function, are remarkable objects that appear in areas as diverse as combinatorics, number theory, statistical mechanics, and electrical engineering. In this article we describe a new surprising application of these objects in graph theory, by showing that the set of all cliques is not forcing for quasirandomness. This provides a natural example of an infinite family of graphs, which is not forcing, and answers a natural question posed by P. Horn.
From Quasirandom graphs to Graph Limits and Graphlets
我们推广了拟随机的概念，它涉及到随机图所满足的一类等价性质。我们证明了图序列在谱距离下的收敛等价于使用（归一化）割距离的收敛。由此产生的图形限制称为graphlets。然后，我们考虑几个家庭的graphlets，特别是，我们的特点graphlets低秩密集和稀疏图。
Let $G$ be a simple graph with maximum degree $Δ(G)$. A subgraph $H$ of $G$ is overfull if $|E(H)|>Δ(G)\lfloor |V(H)|/2 \rfloor$. Chetwynd and Hilton in 1985 conjectured that a graph $G$ on $n$ vertices with $Δ(G)>n/3$ has chromatic index $Δ(G)$ if and only if $G$ contains no overfull subgraph. Glock, Kühn and Osthus in 2016 showed that the conjecture is true for dense quasirandom graphs with even order, and they conjectured that the same should hold for such graphs with odd order. In this paper, we show that the conjecture of Glock, Kühn and Osthus is affirmative.
Forcing Generalized Quasirandom Graphs Efficiently
本文研究广义拟随机图的结构迫使问题，将同态密度所需测试图的顶点数上界改进至 4q^2−q，并在各部分度数互异时改进至 2q+1。
The Poset of Hypergraph Quasirandomness
Chung和Graham在Chungason和Chung-Graham-Wilson关于拟随机图的基本结果之后不久就开始了对k-一致超图拟随机性质的系统研究。在早期关于k-一致超图拟随机性的研究中，一个明显的特征是对图等价的性质对超图不等价，因此超图具有各种不等价的拟随机性质。在过去的二十年里，人们对超图的这些不同的准随机性概念进行了深入的研究，并且出现了一个公开的问题，即确定它们之间的关系。我们的主要结果是确定这些拟随机性质之间的偏序集的影响。这回答了最近的一个问题的钟和继续一个项目开始的钟和格雷厄姆在他们的第一篇论文超图准随机性在20世纪90年代初。
An $n$-vertex graph $G$ of edge density $p$ is considered to be quasirandom if it shares several important properties with the random graph $G(n,p)$. A well-known theorem of Chung, Graham and Wilson states that many such `typical' properties are asymptotically equivalent and, thus, a graph $G$ possessing one such property automatically satisfies the others. In recent years, work in this area has focused on uncovering more quasirandom graph properties and on extending the known results to other discrete structures. In the context of hypergraphs, however, one may consider several different notio…
We employ the absorbing-path method in order to prove two results regarding the emergence of tight Hamilton cycles in the so called {\em two-path} or {\em cherry}-quasirandom $3$-graphs. Our first result asserts that for any fixed real $α>0$, cherry-quasirandom $3$-graphs of sufficiently large order $n$ having minimum $2$-degree at least $α(n-2)$ have a tight Hamilton cycle. Our second result concerns the minimum $1$-degree sufficient for such $3$-graphs to have a tight Hamilton cycle. Roughly speaking, we prove that for every $d,α>0$ satisfying $d + α>1$, any sufficiently large $n$-vertex suc…
Motivated by longstanding conjectures regarding decompositions of graphs into paths and cycles, we prove the following optimal decomposition results for random graphs. Let $0<p<1$ be constant and let $G\sim G_{n,p}$. Let $odd(G)$ be the number of odd degree vertices in $G$. Then a.a.s. the following hold: (i) $G$ can be decomposed into $\lfloorΔ(G)/2\rfloor$ cycles and a matching of size $odd(G)/2$. (ii) $G$ can be decomposed into $\max\{odd(G)/2,\lceilΔ(G)/2\rceil\}$ paths. (iii) $G$ can be decomposed into $\lceilΔ(G)/2\rceil$ linear forests. Each of these bounds is best possible. We actually…
We study forcing pairs for quasirandom graphs. Chung, Graham, and Wilson initiated the study of families $\\mathcal F$ of graphs with the property that if a large graph $G$ has approximately homomorphism density $p^{e(F)}$ for some fixed $p\in(0,1]$ for every $F\in \mathcal F$, then $G$ is quasirandom with density $p$. Such families $\mathcal F$ are said to be forcing. Several forcing families were found over the last three decades and characterising all bipartite graphs $F$ such that $(K_2,F)$ is a forcing pair is a well-known open problem in the area of quasirandom graphs, which is closely r…
Perfect Packings in Quasirandom Hypergraphs II
对于每一个概念的超图quasirandomness已被研究，我们确定了一个大类的超图F，使每一个quasirandom超图H承认一个完美的F-包装。我们关于3-一致超图的一般结果的一个特例的非正式陈述如下。固定一个整数r >= 4和0<p<1。设H是一个n-顶点三元系，|n和以下两个性质：* 对每个满足V（G）=V（H）的图G，G中至少有p部分三角形也是H的边，* 对H的每个顶点x，x的链图是密度至少为p的拟随机图，则H有完美$K_r^{(3)}$ -填充.此外，我们还证明了上述两个假设都不能被削弱，所以在这个意义上，我们的结果是紧的。对于这种特殊情况，一个类似的结论可以通过Keevash的超图爆破引理来证明，对H.
A maximal $\varepsilon$-near perfect matching is a maximal matching which covers at least $(1-\varepsilon)|V(G)|$ vertices. In this paper, we study the number of maximal near perfect matchings in generalized quasirandom and dense graphs. We provide tight lower and upper bounds on the number of $\varepsilon$-near perfect matchings in generalized quasirandom graphs. Moreover, based on these results, we provide a deterministic polynomial time algorithm that for a given dense graph $G$ of order $n$ and a real number $\varepsilon>0$, returns either a conclusion that $G$ has no $\varepsilon$-near pe…
Turán density of cliques of order five in $3$-uniform hypergraphs with quasirandom links
证明所有顶点链接图拟随机且密度大于1/3的3-一致超图必含五顶点团，结果渐近最优。
Eigenvalues and Linear Quasirandom Hypergraphs
设p（k）表示k的分拆函数。对于每个k >= 2，我们描述了一个k-一致超图可以具有的p（k）-1拟随机性质的列表。我们的工作将Kohayakawa-Rödl-Skokan和Conlon-Hàn-Person-Schacht关于线性超图拟随机性的先前概念与Friedman-Wigderson的谱方法联系起来。对于所描述的每一个拟随机性质，我们定义了一个最大和第二大的特征值。我们证明了一个超图满足这些拟随机性质当且仅当它有一个大的谱隙。这就回答了康隆-汉-珀森-沙赫特的一个问题。我们的工作可以看作是Chung-Graham-Wilson关于图的谱结果在超图上的部分推广。
On the Isomorphism Problem of Cayley Graphs of Graph Products
该研究探讨图乘积的Cayley图同构问题，证明顶点群的Cayley图同构可推出对应图乘积的Cayley图同构，为相关同构判定提供理论依据。
In this paper, generalized Cayley graphs are studied. It is proved that every generalized Cayley graph of order 2p is a Cayley graph, where p is a prime. Special attention is given to generalized Cayley graphs on Abelian groups. It is proved that every generalized Cayley graph on an Abelian group with respect to an automorphism which acts as inversion is a Cayley graph if and only if the group is elementary Abelian 2-group, or its Sylow 2-subgroup is cyclic. Necessary and sufficient conditions for a generalized Cayley graph to be unworthy are given.
Mixing (or quasirandom) properties of the natural transition matrix associated to a graph can be quantified by its distance to the complete graph. Different mixing properties correspond to different norms to measure this distance. For dense graphs, two such properties known as spectral expansion and uniformity were shown to be equivalent in seminal 1989 work of Chung, Graham and Wilson. Recently, Conlon and Zhao extended this equivalence to the case of sparse vertex transitive graphs using the famous Grothendieck inequality. Here we generalize these results to the non-commutative, or `quantum'…
The planar cubic Cayley graphs of connectivity 2
我们对连通度为2的平面三次Cayley图进行了分类，为它们中的每一个提供了明确的表示和嵌入。与[9]相结合，这给出了所有平面三次Cayley图的完整描述。
Minimal Cayley graphs with large chromatic number
针对Babai的极小Cayley图问题，构造出色数任意大的有限极小Cayley图
As a main result of this paper we give conditions under which the generalized $X$-join of Cayley graphs is a Cayley graph. In particular, we show that $X$-join of isomorphic Cayley graphs is a Cayley grpah. To do this, new properties for a generalized wreath product of permutation groups are given in the case where the base group acts regularly. These are used to give conditions for the generalized wreath product to contain a regular subgroup, which are then applied to generalized $X$-joins of Cayley graphs. Along the way, it is shown that the generalized $X$-join of isomorphic Cayley graphs w…
In this paper we characterize all of Cayley graphs on dihedral groups with metric dimension two.
Integral Cayley graphs and groups
我们解决了两个公开的问题，某些类别的凯莱图的整数特征值的分类。我们首先分类所有的有限群，有一个“非平凡”的凯莱图整数特征值，从而解决了Abdollahi和Jazaeri提出的问题。凯莱整群的概念是由Klotz和Sander提出的。这些群的每一个凯莱图只有整数特征值。在第二部分中，我们确定了所有的Cayley整群。
Approximating Cayley diagrams versus Cayley graphs
我们构造了一个弱收敛到Cayley图的有限图序列，但是没有标记收敛到相应Cayley图的边。一个类似的构造被用来给出收敛到相同极限的图序列，并且使得其中一个图中的生成树具有不能被另一个图的任何子图逼近的极限。我们给出一个例子，这个子树是一个哈密顿圈，但收敛是指在更强的意义。这些后者涉及到是否有一个哈密顿圈是一个可测试的图形属性。
Two finitely generated monoids are constructed, one finitely presented the other not, whose (directed, unlabelled) Cayley graphs are isomorphic.
In this paper, we consider the problem on the existence of perfect state transfer(PST for short) on semi-Cayley graphs over abelian groups (which are not necessarily regular), i.e on the graphs having semiregular and abelian subgroups of automorphisms with two orbits of equal size. We stablish a characterization of semi-Cayley graphs over abelian groups having PST. As a result, we give a characterization of Cayley graphs over groups with an abelian subgroup of index 2 having PST, which improves the earlier results on Cayley graphs over abelian groups, dihedral groups and dicyclic group and det…
Two-sided Cayley graphs
我们介绍了一个家庭的图形，推广类凯莱图。对于群G的非空子集L，R，双边Cayley图2SC（G;L，R）是顶点集G和一条从x到y的弧的有向图当且仅当y=a^{-1}xb，其中a在L中，b在R中.因此，与凯莱图一样，双侧凯莱图可以用于对网络进行建模，因为可以在每个顶点处实现相同的路由和通信方案。我们确定了双边Cayley图是简单无向图的条件，并给出了它们是连通图、点传递图或Cayley图的充分条件。几个开放的问题提出。给出了许多例子，包括一个12个顶点的连通分量大小为4和8。
On distance two in Cayley graphs of Coxeter groups
考虑Coxeter系统$(W,S)$的Cayley图${\rm C}(W,S)$，并描述了该图中的所有极大$2$ -团，即顶点集中的极大子集，使得任意两个不同元素之间的距离等于$2$ .作为应用，我们证明了Cayley图的每一半的自同构是唯一可扩的，如果$|S|\ge 5$ .
We present a construction that gives an infinite series of divisible design graphs which are Cayley graphs.
Characterising planar Cayley graphs and Cayley complexes in terms of group presentations
我们证明了一个Cayley图可以嵌入到欧氏平面中而不需要顶点的聚点当且仅当它是一个Cayley复形的1-骨架，在去除多余的单形后可以嵌入到平面中。我们还给出了这些凯莱图的特征，根据组表示，并推断出它们可以有效地枚举。
We construct a connected cubic nonnormal Cayley graph on $\mathrm{A}_{2^m-1}$ for each integer $m\geqslant4$ and determine its full automorphism group. This is the first infinite family of connected cubic nonnormal Cayley graphs on nonabelian simple groups.
We prove that every vertex transitive, planar, 1-ended, graph covers every graph whose balls of radius r are isomorphic to the ball of radius r in G for a sufficiently large r. We ask whether this is a general property of finitely presented Cayley graphs, as well as further related questions.
A Note on Eigenvalues of Cayley Graphs
该研究聚焦Cayley图的特征值性质，通过对奇数阶群上的整正规Cayley图进行推导，证明此类非空图必然存在奇特征值，丰富了整图与正规Cayley图的相关理论。
On the Cayleyness of Praeger-Xu graphs
本文研究Praeger-Xu图的Cayley性质，给出了其成为Cayley图的充分必要条件。
A census of Cayley graphs
本文提出方法用于生成阶数有限且包含特定大小Cayley集的群，并列举了有限顶点数下的非同构Cayley图。
On Cayley representations of central Cayley graphs over almost simple groups
本文研究单群与拟单群上中心Cayley图的Cayley表示，证明单群上至多有两个不等价表示，并给出拟单群情形下多项式时间求全部不等价表示的算法。
Automorphisms of Cayley graphs on generalised dicyclic groups
一个图称为GRR，如果它的自同构群规则地作用在它的顶点集上。这样的图必然是凯莱图。Godsil表明，只有两个无限家庭的有限团体不承认GRR：阿贝尔群和广义双循环群。事实上，这样一个群上的任何凯莱图都允许只依赖于群的特定的附加图自同构。最近，多布森和最后两位作者表明，几乎所有的凯莱图阿贝尔群承认没有自同构以外的这些明显的必要。在本文中，我们证明了类似的结果凯莱图的其余家庭的例外群：广义双循环群。
In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph $G(10,2)$, occurring as the graph $40$ in the Foster census of connected symmetric trivalent graphs.
1-Factorizations of Cayley graphs
本文证明了有限群$Q \times H$的所有连通Cayley图都是1-可因子分解的，其中$Q$是任意2-幂阶非平凡群，$H$是任意奇阶群.
Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for the so-called coregular k-uniform hypergraphs. In this paper, we extend this characterization to all k-uniform hypergraphs, not just the coregular ones. Specifically, we prove that if a k-uniform hypergraph satisfies the correct count of a specially defined four-cycle, then there is a gap between its first and second largest eigenvalue.
Extending earlier results of Godsil and of Dobson and Malnic on Johnson graphs, we characterise those merged Johnson graphs $J=J(n,k)_I$ which are Cayley graphs, that is, which are connected and have a group of automorphisms acting regularly on the vertices. We also characterise the merged Johnson graphs which are not Cayley graphs but which have a transitive group of automorphisms with vertex-stabilisers of order $2$. Even though these merged Johnson graphs are all vertex-transitive, we show that only relatively few of them are Cayley graphs or have a transitive group of automorphisms with ve…
Given a sufficiently large and sufficiently dense bipartite graph $G=(A, B; E),$ we present a novel method for decomposing the majority of the edges of $G$ into quasirandom graphs so that the vertex sets of these quasirandom graphs partition the majority of $A.$ The method works for relatively small or sparse graphs, and can be used to substitute the Regularity lemma of Szemerédi in some graph embedding problems.
Normal Edge-Transitive Cayley Graphs of Frobenius Groups
一个群$G$的Cayley图称为正规边传递的，如果它允许$G$的全形（$\operatorname{Sym}(G)$中$G$的正则副本的正规化子）的某个子群的边传递作用.通过对两个素数的乘积为a阶的弗罗贝尼乌斯群的Cayley图的研究，完成了这类图的分类.我们确定这些图的自同构群，特别证明有一个独特的顶点本原的例子，即Fano平面的旗图。
A Cayley graph of a group $H$ is a finite simple graph $Γ$ such that ${\rm Aut}(Γ)$ contains a subgroup isomorphic to $H$ acting regularly on $V(Γ)$, while a Haar graph of $H$ is a finite simple bipartite graph $Σ$ such that ${\rm Aut}(Σ)$ contains a subgroup isomorphic to $H$ acting semiregularly on $V(Σ)$ and the $H$-orbits are equal to the bipartite sets of $Σ$. A Cayley graph is a Haar graph exactly when it is bipartite, but no simple condition is known for a Haar graph to be a Cayley graph. In this paper, we show that the groups $D_6, \, D_8, \, D_{10}$ and $Q_8$ are the only finite inner…
Tight Hamilton Cycles in Linearly Quasirandom 3-Graphs
研究线性拟随机3-图中紧密哈密顿圈问题，给出反例否定特定条件下存在紧密哈密顿圈的猜想，确定p>1/3时渐近尖锐的最小余度阈值，证明相关充分条件，采用吸收法及新连接引理得出结论。
Perfect Packings in Quasirandom Hypergraphs
设k >= 2，F是v个顶点的线性k一致超图。我们证明，如果n足够大，v| n，则每一个具有最小度和常边密度的n阶拟随机k-一致超图$Ω(n^{k-1})$允许完美F-填充. k = 2的情况直接由科姆洛斯、萨尔科齐和塞梅雷迪的爆破引理得出。我们还证明了一些非线性F的积极结果，但在同一时间给出反例，而简单的F是接近线性的。最后，我们解决的情况下，密度趋于零，并证明（在类比图的情况下），稀疏拟随机3-均匀超图承认完美匹配，只要他们的第二大特征值是足够小于的最大特征值。
In this paper we show that two distinct conjectures, the first proposed by Babai and Godsil in $1982$ and the second proposed by Xu in $1998$, concerning the asymptotic enumeration of Cayley graphs are in fact equivalent. This result follows from a more general theorem concerning the asymptotic enumeration of a certain family of Cayley graphs.
Completions of epsilon-dense partial Latin squares; quasirandom k-colorings of graphs
组合数学中的一个经典问题是：给定一个部分拉丁方$P$，什么时候我们可以将$P$完全化为一个拉丁方$L$？本文研究了\textbf{$ε$-dense partial Latin squares}类：\ partial拉丁方，其中每个符号、行和列包含不超过$εn$ -许多非空单元格.基于Nash-Williams的一个猜想，Daykin和Häggkvist证明了所有$\frac{1}{4}$稠密的部分拉丁方都是可补的。在本文中，我们将讨论的证明方法和结果在以前的尝试来解决这个猜想，介绍了一种新的技术，来自Jacobson和Matthews的论文产生随机拉丁方，并使用这种新的技术来研究$ ε$ -密集的部分拉丁方，包含不超过$δn^2$填充细胞总数。特别地，我们构造了所有$ ε$ -稠密部分拉丁方的完备化，它们总共包含不超过$δn^2$填充的单元格，给定$ε< \frac{1}{12}, δ< \frac{ \left(1-12ε\right)^{2}}{10409}$ .特别是，我们证明了所有$9.8 \cdot 10^{-5}$稠密的部分拉丁方是可完成的。我们进一步表明，这样的完成总是可以在多项式时间内找到。这与科尔伯恩的结果形成对比。在第三章中，我们加强了Colbourn的结果，证明了对任意$ε> 0$，完备任意$\left(\frac{1}{2} + ε\right)$ -稠密…
The planar cubic Cayley graphs
我们得到一个完整的描述，平面三次凯莱图，提供了一个明确的介绍和嵌入他们每个人。这是一个富裕的阶级，由几个无限大的家庭组成。我们得到了Mohar，Bonnington和Watkins的证明的反例.我们的分析，使所涉及的图形计算，证实了Droms的猜想。
Cayley graphs and G-graphs of Gyro-groups
本文研究若干陀螺群的凯莱图与G-图结构，证明相关性质，并考察其对称性。
Asymptotic stability of Cayley graphs on abelian groups
本文证明几乎所有有限阿贝尔群的赛雷图都是稳定的，通过研究其常规双覆盖的自同构群最小化问题。
Which finitely generated Abelian groups admit isomorphic Cayley graphs?
我们证明了非正则生成的阿贝尔群的Cayley图是刚性的.作为结果，我们得到两个生成的阿贝尔群允许同构Cayley图当且仅当它们具有相同的秩和它们的挠部分具有相同的基数。证明只使用基本参数，并制定了几何语言。
Sidorenko's Conjecture asserts that every bipartite graph H has the Sidorenko property, i.e., a quasirandom graph minimizes the density of H among all graphs with the same edge density. We study a stronger property, which requires that a quasirandom multipartite graph minimizes the density of H among all graphs with the same edge densities between its parts; this property is called the step Sidorenko property. We show that many bipartite graphs fail to have the step Sidorenko property and use our results to show the existence of a bipartite edge-transitive graph that is not weakly norming; thi…
The purpose of this note is to prove that the set of connective constants of Cayley graphs contains a Cantor space.
We construct a 2-generated group $Γ$ such that its Cayley graph possesses finite connected subsets with arbitrarily big finite Heesch number.
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