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Spectral characterization of the complete graph removing a cycle 去除一个周期的完整图谱特征
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-02-09 DOI: 10.1016/j.jcta.2024.105868
Muhuo Liu , Xiaofeng Gu , Haiying Shan , Zoran Stanić

A graph is determined by its spectrum if there is not another graph with the same spectrum. Cámara and Haemers proved that the graph KnCk, obtained from the complete graph Kn with n vertices by deleting all edges of a cycle Ck with k vertices, is determined by its spectrum for k{3,4,5}, but not for k=6. In this paper, we show that k=6 is the unique exception for the spectral determination of KnCk.

如果不存在另一个具有相同频谱的图,则该图由其频谱决定。Cámara 和 Haemers 证明,通过删除具有 k 个顶点的循环 Ck 的所有边,从具有 n 个顶点的完整图 Kn 得到的图 Kn∖Ck,在 k∈{3,4,5} 时由其谱决定,但在 k=6 时则不是。在本文中,我们将证明 k=6 是 Kn∖Ck 的谱确定性的唯一例外。
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引用次数: 0
The divisor class group of a discrete polymatroid 离散多面体的因子类群
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-02-08 DOI: 10.1016/j.jcta.2024.105869
Jürgen Herzog , Takayuki Hibi , Somayeh Moradi , Ayesha Asloob Qureshi

In this paper we introduce toric rings of multicomplexes. We show how to compute the divisor class group and the class of the canonical module when the toric ring is normal. In the special case that the multicomplex is a discrete polymatroid, its toric ring is studied deeply for several classes of polymatroids.

本文介绍了多复数的环。我们展示了如何计算环是正态时的除数类群和典型模块类。在多面体是离散多面体的特殊情况下,我们深入研究了几类多面体的环。
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引用次数: 0
Large sum-free sets in Z5n Z5n 中的大无和集
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-02-02 DOI: 10.1016/j.jcta.2024.105865
Vsevolod F. Lev

It is well-known that for a prime p2(mod3) and integer n1, the maximum possible size of a sum-free subset of the elementary abelian group Zpn is 13(p+1)pn1. However, the matching stability result is known for p=2 only. We consider the first open case p=5 showing that if AZ5n is a sum-free subset with |A|>325n1, then there are a subgroup H<Z5n of size |H|=5n1 and an element eH such that A(e+H)(e+H).

众所周知,对于素数 p≡2(mod3)和整数 n≥1,初等无方组 Zpn 的无和子集的最大可能大小为 13(p+1)pn-1。然而,已知的匹配稳定性结果仅适用于 p=2。我们考虑第一种开放情况 p=5 表明,如果 A⊆Z5n 是|A|>32⋅5n-1 的无和子集,那么存在大小为 |H|=5n-1的子群 H<Z5n 和元素 e∉H,使得 A⊆(e+H)∪(-e+H)。
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引用次数: 0
Block-transitive 2-designs with a chain of imprimitive point-partitions 带有一连串隐含点分区的块变换 2 设计
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-02-01 DOI: 10.1016/j.jcta.2024.105866
Carmen Amarra , Alice Devillers , Cheryl E. Praeger

More than 30 years ago, Delandtsheer and Doyen showed that the automorphism group of a block-transitive 2-design, with blocks of size k, could leave invariant a nontrivial point-partition, but only if the number of points was bounded in terms of k. Since then examples have been found where there are two nontrivial point partitions, either forming a chain of partitions, or forming a grid structure on the point set. We show, by construction of infinite families of designs, that there is no limit on the length of a chain of invariant point partitions for a block-transitive 2-design. We introduce the notion of an ‘array’ of a set of points which describes how the set interacts with parts of the various partitions, and we obtain necessary and sufficient conditions in terms of the ‘array’ of a point set, relative to a partition chain, for it to be a block of such a design.

30 多年前,德兰切尔和多延证明,大小为 k 块的块变换 2 设计的自动形群可以使一个非难点分区保持不变,但前提是点的数量以 k 为界。自那时起,人们发现了有两个非难点分区的例子,它们要么形成了分区链,要么在点集中形成了网格结构。我们通过构建无穷设计族证明,对于块过渡 2 设计,不变点分区链的长度没有限制。我们引入了点集 "阵列 "的概念,它描述了点集如何与不同分区的部分相互作用,我们还获得了点集 "阵列 "相对于分区链的必要条件和充分条件,从而使其成为这种设计的区块。
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引用次数: 0
Large monochromatic components in colorings of complete hypergraphs 完整超图着色中的大型单色成分
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-02-01 DOI: 10.1016/j.jcta.2024.105867
Lyuben Lichev , Sammy Luo

Gyárfás famously showed that in every r-coloring of the edges of the complete graph Kn, there is a monochromatic connected component with at least nr1 vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for k-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of 1+o(1) as the number of colors grows. We further generalize these questions to ask about counts of vertex s-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.

Gyárfás 的著名研究表明,在完整图 Kn 的边的每 r 种着色中,都存在一个至少有 nr-1 个顶点的单色连通部分。最近,Conlon、Tyomkyn 和第二位作者的一项研究解决了具有许多边的单色连通成分的类似问题。在本文中,我们研究了这些问题在 k-uniform 超图中的推广。在将连通性定义扩展到更高均匀性的广泛范围内,我们为最大单色成分的大小提供了上界和下界,随着颜色数量的增加,上界和下界的紧密程度可达 1+o(1)。我们进一步将这些问题推广到大型单色分量边缘中包含的顶点 s 集的计数。最后,我们将针对两种颜色的特殊情况给出更精确的结果。
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引用次数: 0
A study on free roots of Borcherds-Kac-Moody Lie superalgebras 关于 Borcherds-Kac-Moody Lie 超代数自由根的研究
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-01-25 DOI: 10.1016/j.jcta.2024.105862
Shushma Rani , G. Arunkumar

Consider a Borcherds-Kac-Moody Lie superalgebra, denoted as g, associated with the graph G. This Lie superalgebra is constructed from a free Lie superalgebra by introducing three sets of relations on its generators: (1) Chevalley relations, (2) Serre relations, and (3) The commutation relations derived from the graph G.

The Chevalley relations lead to a triangular decomposition of g as g=n+hn, where each root space gα is contained in either n+ or n. Importantly, each gα is determined solely by relations (2) and (3). This paper focuses on the root spaces of g that are unaffected by the Serre relations. We refer to these root spaces as “free roots” of g (these root spaces are free from the Serre relations and can be associated with certain grade spaces of freely partially commutative Lie superalgebras, as detailed in Lemma 3.10. Consequently, we refer to them as “free roots,” and the corresponding root spaces in g as “free root spaces” [cf. Definition 2.6]). Since these root spaces only involve commutation relations derived from the graph G, we can examine them purely from a combinatorial perspective.

We employ heaps of pieces to analyze these root spaces and establish various combinatorial properties. We develop two distinct bases for these root spaces of g: We extend Lalonde's Lyndon heap basis, originally designed for free partially commutative Lie algebras, to accommodate free partially commutative Lie superalgebras. We expand upon the basis introduced in the reference [1], designed for the free root spaces of Borcherds algebras, to encompass BKM superalgebras. This extension is achieved by investigating the combinatorial properties of super Lyndon heaps. Additionally, we also explore several other combinatorial properties related to free roots.

考虑一个与图 G 相关联的 Borcherds-Kac-Moody Lie 上代数,记为 g。这个 Lie 上代数是通过在其生成器上引入三组关系从自由 Lie 上代数构造而成的:(1) 切瓦利关系;(2) 塞雷关系;(3) 由图 G 导出的换向关系。切瓦利关系导致 g 的三角分解为 g=n+⊕h⊕n-,其中每个根空间 gα 都包含在 n+ 或 n- 中。重要的是,每个 gα 完全由关系式 (2) 和 (3) 决定。本文重点讨论不受塞尔关系影响的 g 的根空间。我们把这些根空间称为 g 的 "自由根"(这些根空间不受塞尔关系的影响,可以与自由部分交换 Lie 超的某些级数空间相关联,详见定理 3.10)。因此,我们把它们称为 "自由根",把 g 中相应的根空间称为 "自由根空间"[参见定义 2.6]。由于这些根空间只涉及从图 G 派生的换元关系,我们可以纯粹从组合的角度来研究它们。我们为 g 的这些根空间开发了两种不同的基础:我们扩展了拉隆德的林顿堆基础(Lyndon heap basis),该基础最初是为自由部分换元李代数设计的,现在也适用于自由部分换元李超代数。我们扩展了参考文献[1]中引入的基础,该基础是为博彻兹代数的自由根空间设计的,以涵盖 BKM 超。这一扩展是通过研究超林顿堆的组合性质实现的。此外,我们还探讨了与自由根相关的其他一些组合性质。
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引用次数: 0
Further refinements of Wilf-equivalence for patterns of length 4 长度为 4 的图案的 Wilf 等价性的进一步完善
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-01-25 DOI: 10.1016/j.jcta.2024.105863
Robin D.P. Zhou , Yongchun Zang , Sherry H.F. Yan

In this paper, we construct a bijection between 3142-avoiding permutations and 3241-avoiding permutations which proves the equidistribution of five classical set-valued statistics. Our bijection also enables us to establish a bijection between 3142-avoiding permutations and 4132-avoiding permutations, and a bijection between 2413-avoiding permutations and 1423-avoiding permutations, both of which preserve five classical set-valued statistics. Our results are generalizations of several conjectures posed by Burstein.

在本文中,我们构建了 3142 避开排列和 3241 避开排列之间的偏射,证明了五个经典的集值统计的等分布。我们的偏射还使我们能够建立 3142 避开排列与 4132- 避开排列之间的偏射,以及 2413-avoiding permutations 与 1423-avoiding permutations 之间的偏射,这两个偏射都保留了五个经典的集值统计量。我们的结果是对布尔斯坦提出的几个猜想的概括。
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引用次数: 0
New results on orthogonal arrays OA(3,5,4n + 2) 关于正交阵列 OA(3,5,4n + 2)的新成果
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-01-24 DOI: 10.1016/j.jcta.2024.105864
Dongliang Li, Haitao Cao

An orthogonal array of index unity, order v, degree 5 and strength 3, or an OA(3,5,v) in short, is a 5×v3 array on v symbols and in every 3×v3 subarray, each 3-tuple column vector occurs exactly once. The existence of an OA(3,5,4n+2) is still open except for few known infinite classes of n. In this paper, we introduce a new combinatorial structure called three dimensions orthogonal complete large sets of disjoint incomplete Latin squares and use it to obtain many new infinite classes of OA(3,5,4n+2)s.

索引为 unity、阶数为 v、阶数为 5、强度为 3 的正交数组,简称 OA(3,5,v),是关于 v 个符号的 5×v3 数组,在每个 3×v3 子数组中,每个 3 元组列向量恰好出现一次。本文引入了一种新的组合结构,称为三维正交完整大集不完全拉丁正方形,并利用它得到了许多新的无穷类 OA(3,5,4n+2)。
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引用次数: 0
q-Supercongruences from Jackson's ϕ78 summation and Watson's ϕ78 transformation 从杰克逊的ϕ78求和与沃森的ϕ78变换中得出的q-超级共轭关系
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2024-01-12 DOI: 10.1016/j.jcta.2023.105853
Chuanan Wei

q-Supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial are very rare in the literature. In this paper, we establish some q-supercongruences modulo the fifth and sixth powers of a cyclotomic polynomial in terms of Jackson's ϕ78 summation, Watson's ϕ78 transformation, the creative microscoping method recently introduced by Guo and Zudilin, and the Chinese remainder theorem for coprime polynomials. More concretely, we give a q-analogue of a nice formula due to Long and Ramakrishna [Adv. Math. 290 (2016), 773–808] and two q-supercongruences involving double series.

环状多项式的五次幂和六次幂的 q 上共轭在文献中非常罕见。在本文中,我们根据杰克逊的ϕ78求和、沃森的ϕ78变换、郭和祖迪林最近提出的创造性微分法以及中国余数定理,建立了一些环状多项式的五次和六次幂的 q 次共轭。更具体地说,我们给出了 Long 和 Ramakrishna [Adv. Math. 290 (2016), 773-808] 提出的一个漂亮公式的 q-analogue 以及两个涉及双数列的 q-supercongruences 。
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引用次数: 0
Nowhere-zero 3-flows in Cayley graphs on supersolvable groups 超可溶群上 Cayley 图中的无处零 3 流
IF 1.1 2区 数学 Q2 MATHEMATICS Pub Date : 2023-12-28 DOI: 10.1016/j.jcta.2023.105852
Junyang Zhang , Sanming Zhou

Tutte's 3-flow conjecture asserts that every 4-edge-connected graph admits a nowhere-zero 3-flow. We prove that this conjecture is true for every Cayley graph of valency at least four on any supersolvable group with a noncyclic Sylow 2-subgroup and every Cayley graph of valency at least four on any group whose derived subgroup is of square-free order.

Tutte 的 3 流猜想断言,每个 4 边连接图都有一个无处为零的 3 流。我们证明,对于任何具有非循环 Sylow 2 子群的可超溶群上每一个至少四价的 Cayley 图,以及任何派生子群为无平方阶的群上每一个至少四价的 Cayley 图,这一猜想都是真的。
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引用次数: 0
期刊
Journal of Combinatorial Theory Series A
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