Discrete Mathematics最新文献

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Fibonacci and Catalan paths in a wall 墙上的斐波纳契和加泰罗尼亚路径

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-24 DOI: 10.1016/j.disc.2024.114268

Jean-Luc Baril , José L. Ramírez

We study the distribution of some statistics (width, number of steps, length, area) defined for paths contained in walls. We present the results by giving generating functions, asymptotic approximations, as well as some closed formulas. We prove algebraically that paths in walls of a given width and ending on the x-axis are enumerated by the Catalan numbers, and we provide a bijection between these paths and Dyck paths. We also find that paths in walls with a given number of steps are enumerated by the Fibonacci numbers. Finally, we give a constructive bijection between the paths in walls of a given length and peakless Motzkin paths of the same length.

我们研究了为墙内路径定义的一些统计量（宽度、步数、长度、面积）的分布。我们通过给出生成函数、渐近近似值以及一些封闭公式来呈现结果。我们用代数方法证明，在给定宽度的墙壁中，以 x 轴为终点的路径可以用加泰罗尼亚数枚举，并提供了这些路径与戴克路径之间的双射关系。我们还发现，具有给定步数的墙内路径可以用斐波那契数枚举。最后，我们给出了给定长度的墙内路径与相同长度的无峰莫兹金路径之间的构造偏射。

引用次数: 0

On the inclusion chromatic index of a Halin graph 论哈林图的包含色度指数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-23 DOI: 10.1016/j.disc.2024.114266

Ningge Huang, Yi Tan, Lily Chen

An inclusion-free edge-coloring of a graph G with

δ (G) \geq 2

is a proper edge-coloring such that the set of colors incident with any vertex is not contained in the set of colors incident to any of its neighbors. The minimum number of colors needed in an inclusion-free edge-coloring of G is called the

i n c l u s i o n

f r e e

c h r o m a t i c i n d e x

, denoted by

χ_{\subset}^{'} (G)

. In this paper, we show that for a Halin graph G with maximum degree

Δ \geq 4

, if G is isomorphic to a wheel

W_{Δ + 1}

where Δ is odd, then

χ_{\subset}^{'} (G) = Δ + 2

, otherwise

χ_{\subset}^{'} (G) = Δ + 1

. We also show a special cubic Halin graph with

χ_{\subset}^{'} (G) = 5

δ(G)≥2的图 G 的无包含边着色是一种适当的边着色，使得任何顶点的颜色集合都不包含在其任何相邻顶点的颜色集合中。G 的无包含边染色所需的最少颜色数称为无包含色度指数，用 χ⊂′(G)表示。在本文中，我们证明了对于最大度数为 Δ≥4 的 Halin 图 G，如果 G 与 Δ 为奇数的轮 WΔ+1 同构，则 χ⊂′(G)=Δ+2 ，否则 χ⊂′(G)=Δ+1。我们还展示了一个特殊的立方哈林图，其χ⊂′(G)=5。

引用次数: 0

Cyclic sieving and dihedral sieving on noncrossing (1,2)-configurations 非交叉 (1,2) 构型上的循环筛分和二面筛分

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.disc.2024.114262

Chuyi Zeng, Shiwen Zhang

<div>Verifying a suspicion of Propp and Reiner concerning the cyclic sieving phenomenon (CSP), M. Thiel introduced a Catalan object called noncrossing <math><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>)</mo></math>-configurations (denoted by <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math>), which is a class of set partitions of <math><mo>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>]</mo></math>. More precisely, Thiel proved that, with a natural action of the cyclic group <math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></math> on <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, the triple <math><mo>(</mo><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>)</mo></math> exhibits the CSP, where <math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo><mo>≔</mo><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mo>[</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></mrow></mfrac><msub><mrow><mo>[</mo><mtable><mtr><mtd><mn>2</mn><mi>n</mi></mtd></mtr><mtr><mtd><mi>n</mi></mtd></mtr></mtable><mo>]</mo></mrow><mrow><mi>q</mi></mrow></msub></math> is MacMahon's q-Catalan number. Recently, in a study of the fermionic diagonal coinvariant ring <math><mi>F</mi><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, Jesse Kim found a combinatorial basis for <math><mi>F</mi><mi>D</mi><msub><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msub></math> indexed by <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math>. In this paper, we continue to study <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math> and obtain the following results:<ul><li>(1)We define a statistic on <math><msub><mrow><mi>X</mi></mrow><mrow><mi>n</mi></mrow></msub></math> whose generating function is <math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math>, which answers a problem of Thiel.</li><li>(2)We show that <math><msub><mrow><mtext>Cat</mtext></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>q</mi><mo>)</mo></math> is equivalent to<math><munder><mo>∑</mo><mrow><mtable><mtr><mtd><mi>k</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mtd></mtr><mtr><mtd><mn>2</mn><mi>k</mi><mo>+</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mtd></mtr></mtable></mrow></munder><msub><mrow><mo>[</mo><mtable><mtr><mtd><mi>n</mi><mo>−</mo><mn

为了验证普罗普和赖纳关于循环筛分现象（CSP）的猜想，蒂尔（M. Thiel）引入了一个称为非交叉（1,2）配置（用 Xn 表示）的加泰罗尼亚对象，它是 [n-1] 的一类集合分区。更确切地说，蒂尔证明了在循环群 Cn-1 对 Xn 的自然作用下，三重（Xn,Cn-1,Catn(q)）展示了 CSP，其中 Catn(q)≔1[n+1]q[2nn]q 是麦克马洪的 q 加泰罗尼亚数。最近，在对费米对角共变环 FDRn 的研究中，Jesse Kim 发现了以 Xn 为索引的 FDRn 组合基。(2)我们证明 Catn(q) 等价于∑k,x,y2k+x+y=n-1[n-12k,x,y]qCatk(q)qk+(x2)+(y2)+(n2) modulo qn-1-1，这回答了 Jesse Kim 的一个问题。(3)我们考虑二面筛分，它是 CSP 的广义化，最近由 Rao 和 Suk 提出。在二面群 I2(n-1)（偶数 n）的自然作用下，我们证明了 Xn 上的二面筛分结果。

引用次数: 0

Graphs with the minimum spectral radius for given independence number 给定独立数时具有最小谱半径的图形

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-20 DOI: 10.1016/j.disc.2024.114265

Yarong Hu , Qiongxiang Huang , Zhenzhen Lou

Let $G_{n, α}$ be the set of connected graphs with order n and independence number α. The graph with the minimum spectral radius among $G_{n, α}$ is called the minimizer graph. Stevanović in the classical book [Spectral Radius of Graphs, Academic Press, Amsterdam, 2015] pointed out that determining the minimizer graph in $G_{n, α}$ appears to be a tough problem. Recently, Lou and Guo (2022) [14] proved that the minimizer graph in $G_{n, α}$ must be a tree if $α \geq ⌈ \frac{n}{2} ⌉$ . In this paper, we further give the structural features for the minimizer graph in detail, and then provide a constructing theorem for it. Thus, theoretically we determine the minimizer graphs in $G_{n, α}$ along with their spectral radius for any given $α \geq ⌈ \frac{n}{2} ⌉$ . As an application, we determine all the minimizer graphs in $G_{n, α}$ for $α = n - 5, n - 6$ along with their spectral radius.

设 Gn,α 是阶数为 n 且独立数为 α 的连通图集合，Gn,α 中谱半径最小的图称为最小图。Stevanović 在其经典著作[Spectral Radius of Graphs, Academic Press, Amsterdam, 2015]中指出，确定 Gn,α 中的最小图似乎是一个难题。最近，Lou 和 Guo（2022）[14] 证明，如果α≥⌈n2⌉，则 Gn,α 中的最小图一定是树。本文进一步详细给出了最小图的结构特征，并给出了其构造定理。因此，我们从理论上确定了 Gn,α 中的最小化图，以及任意给定 α≥⌈n2⌉ 时它们的谱半径。作为应用，我们确定了 α=n-5,n-6 时 Gn,α 中的所有最小图及其谱半径。

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引用次数: 0

A note on rainbow stackings of random edge-colorings of hypergraphs 关于超图随机边着色的彩虹堆积的说明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.disc.2024.114261

Ran Gu

A rainbow stacking of r-edge-colorings $χ_{1}, \dots, χ_{m}$ of the complete d-uniform hypergraph on n vertices is a way of superimposing $χ_{1}, \dots, χ_{m}$ so that no edges of the same color are superimposed on each other. The definition of rainbow stackings of graphs was proposed by Alon, Defant, and Kravitz, and they determined a sharp threshold for r (as a function of m and n) governing the existence and nonexistence of rainbow stackings of random r-edge-colorings $χ_{1}, \dots, χ_{m}$ of the complete graph $K_{n}$ . In this paper, we extend their result to d-uniform hypergraph, obtain a sharp threshold for r controlling the existence and nonexistence of rainbow stackings of random r-edge-colorings $χ_{1}, \dots, χ_{m}$ of the complete d-uniform hypergraph for $d \geq 3$ .

n 个顶点上的完整 d-Uniform 超图的 r 边颜色 χ1，...，χm 的彩虹叠加是一种叠加 χ1，...，χm 的方法，这样就不会有相同颜色的边相互叠加。图的彩虹叠加定义是由 Alon、Defant 和 Kravitz 提出的，他们确定了 r 的一个尖锐临界值（作为 m 和 n 的函数），该临界值决定了完整图 Kn 的随机 r 边颜色χ1,...,χm 的彩虹叠加存在与否。在本文中，我们将他们的结果推广到 d-uniform hypergraph，得到了一个控制完整 d-uniform hypergraph 的随机 r 边着色 χ1,...,χm 的彩虹堆叠存在与否的 r 的尖锐阈值，且 d≥3 时。

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引用次数: 0

Group divisible designs with block size 4 and group sizes 4 and 10 and some other 4-GDDs 组块大小为 4、组块大小为 4 和 10 的可分设计，以及其他一些 4-GDD 设计

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.disc.2024.114254

Changyuan Wang , R. Julian R. Abel , Thomas Britz , Yudhistira A. Bunjamin , Diana Combe

In this paper, we consider the existence of group divisible designs (GDDs) with block size 4 and group sizes 4 and 10. We show that a 4-GDD of type $4^{t} 10^{s}$ exists when the necessary conditions are satisfied, except possibly for a finite number of cases with $4 t + 10 s \leq 178$ . We also give some new examples of 4-GDDs for which the number of points is 51, 54 or some value less than or equal to 50.

在本文中，我们考虑了块大小为 4、组大小为 4 和 10 的组可分割设计（GDD）的存在性。我们证明，在满足必要条件的情况下，4t10s 类型的 4-GDD 是存在的，4t+10s≤178 的有限个例除外。我们还给出了一些点数为 51、54 或小于等于 50 的 4-GDD 的新例子。

引用次数: 0

Critically 3-frustrated signed graphs 关键的 3 个沮丧的签名图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-19 DOI: 10.1016/j.disc.2024.114258

Chiara Cappello , Reza Naserasr , Eckhard Steffen , Zhouningxin Wang

Extending the notion of maxcut, the study of the frustration index of signed graphs is one of the basic questions in the theory of signed graphs. Recently two of the authors initiated the study of critically frustrated signed graphs. That is a signed graph whose frustration index decreases with the removal of any edge. The main focus of this study is on critical signed graphs which are not edge-disjoint unions of critically frustrated signed graphs (namely indecomposable signed graphs) and which are not built from other critically frustrated signed graphs by subdivision. We conjecture that for any given k there are only finitely many critically k-frustrated signed graphs of this kind.

Providing support for this conjecture we show that there are only two of such critically 3-frustrated signed graphs where there is no pair of edge-disjoint negative cycles. Similarly, we show that there are exactly ten critically 3-frustrated signed planar graphs that are neither decomposable nor subdivisions of other critically frustrated signed graphs. We present a method for building indecomposable critically frustrated signed graphs based on two given such signed graphs. We also show that the condition of being indecomposable is necessary for our conjecture.

从 maxcut 的概念出发，研究有符号图的挫折指数是有符号图理论的基本问题之一。最近，两位作者发起了对临界挫折有符号图的研究。这是一种有符号图，其沮度指数随着任何边的移除而减小。本研究的重点是临界有符号图，这些图不是临界受挫有符号图（即不可分解有符号图）的边缘相交的联合体，也不是通过细分从其他临界受挫有符号图建立起来的。我们猜想，对于任何给定的 k，只有有限多个此类临界 k 受挫有符号图。为了支持这一猜想，我们证明了只有两个此类临界 3 受挫有符号图不存在一对边缘相接的负循环。同样，我们证明了正好有十个临界三挫折有符号平面图既不是可分解的，也不是其他临界三挫折有符号图的细分。我们提出了一种基于两个给定的此类有符号图形构建不可分解的临界受挫有符号图形的方法。我们还证明了不可分解的条件对于我们的猜想是必要的。

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引用次数: 0

Distance-regular graphs with classical parameters that support a uniform structure: Case q ≥ 2 具有支持统一结构的经典参数的距离规则图：情况 q ≥ 2

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-18 DOI: 10.1016/j.disc.2024.114263

Blas Fernández , Roghayeh Maleki , Štefko Miklavič , Giusy Monzillo

Let $Γ = (X, R)$ denote a finite, simple, connected, and undirected non-bipartite graph with vertex set X and edge set $R$ . Fix a vertex $x \in X$ , and define $R_{f} = R ∖ {y z | \partial (x, y) = \partial (x, z)}$ , where ∂ denotes the path-length distance in Γ. Observe that the graph $Γ_{f} = (X, R_{f})$ is bipartite. We say that Γ supports a uniform structure with respect to x whenever $Γ_{f}$ has a uniform structure with respect to x in the sense of Miklavič and Terwilliger [7].

Assume that Γ is a distance-regular graph with classical parameters $(D, q, α, β)$ and diameter $D \geq 4$ . Recall that q is an integer such that $q \notin {- 1, 0}$ . The purpose of this paper is to study when Γ supports a uniform structure with respect to x. We studied the case $q \leq 1$ in [3], and so in this paper we assume $q \geq 2$ . Let $T = T (x)$ denote the Terwilliger algebra of Γ with respect to x. Under an additional assumption that every irreducible T-module with endpoint 1 is thin, we show that if Γ supports a uniform structure with respect to x, then either $α = 0$ or $α = q$ , $β = q^{2} (q^{D} - 1) / (q - 1)$ , and $D \equiv 0 (\mod 6)$ .

让Γ=(X,R) 表示具有顶点集 X 和边集 R 的有限、简单、连通和不定向的非双向图。固定一个顶点 x∈X，定义 Rf=R∖{yz|∂(x,y)=∂(x,z)}，其中∂表示Γ中的路径长度距离。请注意，图 Γf=(X,Rf) 是双向的。假设 Γ 是一个距离规则图，其经典参数为 (D,q,α,β)，直径为 D≥4。回顾一下，q 是一个整数，使得 q∉{-1,0}。我们在 [3] 中研究过 q≤1 的情况，因此本文假设 q≥2 。让 T=T(x) 表示 Γ 关于 x 的泰尔维利格代数。在每个端点为 1 的不可还原 T 模块都是薄的这一额外假设下，我们证明了如果 Γ 支持关于 x 的均匀结构，那么要么 α=0 要么 α=q，β=q2(qD-1)/(q-1)，D≡0(mod6)。

引用次数: 0

New results on asymmetric orthogonal arrays with strength t ≥ 3 强度 t ≥ 3 的非对称正交阵列的新结果

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-16 DOI: 10.1016/j.disc.2024.114264

Xiaodong Niu , Guangzhou Chen , Qiang Gao , Shanqi Pang

The orthogonal array holds significant importance as a research topic within the realms of combinatorial design theory and experimental design theory, with widespread applications in statistics, computer science, coding theory and cryptography. This paper presents three constructions for asymmetric orthogonal arrays including juxtaposition, generator matrices over Galois fields and mixed difference matrices. Subsequently, many new infinite families of asymmetric orthogonal arrays with strength $t \geq 3$ are obtained. Furthermore, some new infinite families of large sets of orthogonal arrays with mixed levels are also obtained.

正交阵列作为组合设计理论和实验设计理论领域的一个重要研究课题，在统计学、计算机科学、编码理论和密码学中有着广泛的应用。本文介绍了非对称正交阵列的三种构造，包括并列、伽罗瓦域上的生成矩阵和混合差分矩阵。随后，得到了强度 t≥3 的许多新的非对称正交阵列无穷族。此外，还得到了一些新的具有混合水平的大集正交阵列无穷族。

引用次数: 0

Corrigendum to “On the sum of the first two largest signless Laplacian eigenvalues of a graph” [Discrete Math. 347 (2024) 114035] 对 "论图的前两个最大无符号拉普拉奇特征值之和 "的更正 [Discrete Math. 347 (2024) 114035]

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-09-16 DOI: 10.1016/j.disc.2024.114241

Zhi-Bin Du , Zi-Ming Zhou , Hai-Ying Shan , Chang-Xiang He

引用次数: 0

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