Discrete Mathematics最新文献_第10页

A Note on Hamiltonian-intersecting families of graphs 关于哈密顿相交图族的说明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-07-15 DOI: 10.1016/j.disc.2024.114160

How many graphs on an n-point set can we find such that any two have connected intersection? Berger, Berkowitz, Devlin, Doppelt, Durham, Murthy and Vemuri showed that the maximum is exactly $1 / 2^{n - 1}$ of all graphs. Our aim in this short note is to give a ‘directed’ version of this result; we show that a family of oriented graphs such that any two have strongly-connected intersection has size at most $1 / 3^{n}$ of all oriented graphs. We also show that a family of graphs such that any two have Hamiltonian intersection has size at most $1 / 2^{n}$ of all graphs, verifying a conjecture of the above authors.

在一个 n 点集合上，我们能找到多少个任意两个有相连交集的图形？Berger、Berkowitz、Devlin、Doppelt、Durham、Murthy 和 Vemuri 的研究表明，最大值恰好是所有图形的 1/2n-1 。我们在这篇短文中的目的是给出这一结果的 "有向 "版本；我们证明了这样一个有向图族，即任意两个有强连接交集的有向图的大小最多为所有有向图的 1/3n。我们还证明，任意两个具有哈密顿交集的图族的大小最多为所有图的 1/2n ，这验证了上述作者的猜想。

引用次数: 0

Some determinantal representations of derangement polynomials of types B and D B 型和 D 型出差多项式的一些行列式表示法

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-07-14 DOI: 10.1016/j.disc.2024.114155

Chak-On Chow

Chow (2024) recently computed expressions of the types B and D derangement polynomials $d_{n}^{B} (q) = \sum_{σ \in D_{n}^{B}} q^{fmaj (σ)}$ and $d_{n}^{D} (q) = \sum_{σ \in D_{n}^{D}} q^{maj (σ)}$ as tridiagonal and lower Hessenberg determinants of order n. Qi, Wang, and Guo (2016), based on a determinantal formula for the nth derivative of a quotient of two functions, derived an expression of the classical derangement number $d_{n} = n! \sum_{k = 0}^{n} {(- 1)}^{k} / k!$ as a tridiagonal determinant of order $n + 1$ . By q-extending the approach of Qi et al., we present in this work yet another determinantal expressions of $d_{n}^{B} (q)$ and $d_{n}^{D} (q)$ as determinants of order $n + 1$ .

Chow (2024) 最近计算了 B 和 D 型失真多项式 dnB(q)=∑σ∈DnBqfmaj(σ) 和 dnD(q)=∑σ∈DnDqmaj(σ) 的表达式，它们是 n 阶的三对角和下海森伯行列式。Qi、Wang和Guo（2016）基于两个函数商的n次导数的行列式公式，推导出了经典失真数dn=n！∑k=0n(-1)k/k！作为n+1阶的三对角行列式的表达式。通过对齐等人的方法进行 q 扩展，我们在本文中提出了 dnB(q) 和 dnD(q) 作为 n+1 阶行列式的另一种行列式表达式。

引用次数: 0

On a property of 2-connected graphs and Dirac's Theorem 关于 2 连接图的一个属性和狄拉克定理

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-07-14 DOI: 10.1016/j.disc.2024.114153

Alexandr Kostochka , Ruth Luo , Grace McCourt

We refine a property of 2-connected graphs described in the classical paper of Dirac from 1952 and use the refined property to somewhat shorten Dirac's proof of the fact that each 2-connected n-vertex graph with minimum degree at least k has a cycle of length at least $\min {n, 2 k}$ .

我们完善了 1952 年狄拉克经典论文中描述的 2 连接图的一个性质，并利用这一完善的性质在一定程度上缩短了狄拉克对以下事实的证明：每个最小度至少为 k 的 2 连接 n 顶点图至少有一个长度为 min{n,2k} 的循环。

引用次数: 0

A class of BCH codes of length 2(q2m−1)q+1 and their duals 一类长度为 2(q2m-1)q+1 的 BCH 码及其对偶码

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-07-14 DOI: 10.1016/j.disc.2024.114152

Huan Zhu, Jin Li, Shixin Zhu

BCH codes are a special subclass of cyclic codes. In many cases, BCH codes are best cyclic codes and they have wide applications in communication and storage systems. In this paper, we investigate the parameters of a class of narrow-sense BCH codes over $G F (q)$ of length $\frac{2 (q^{2 m} - 1)}{q + 1}$ with small and large dimensions. We study the q-cyclotomic cosets modulo $\frac{2 (q^{2 m} - 1)}{q + 1}$ , determine the dimensions of these BCH codes and give the lower bounds on their minimum distances. Furthermore, we present the lower bounds on the minimum distances of their dual codes.

BCH 码是循环码的一个特殊子类。在许多情况下，BCH 码是最好的循环码，它们在通信和存储系统中有着广泛的应用。在本文中，我们研究了一类长度为 2(q2m-1)q+1 的 GF(q) 上的小维度和大维度窄义 BCH 码的参数。我们研究了 2(q2m-1)q+1 modulo 的 q-cyclotomic cosets，确定了这些 BCH 码的维数，并给出了它们的最小距离下界。此外，我们还给出了它们对偶码的最小距离下界。

引用次数: 0

Spectral extrema of graphs with fixed size: Forbidden triangles and pentagons 具有固定大小的图形的谱极值：禁止三角形和五边形

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-07-10 DOI: 10.1016/j.disc.2024.114151

Shuchao Li , Yuantian Yu

Nosal (1970) and Nikiforov (2002) showed that if graph G is $C_{3}$ -free of size m, then the spectral radius of G satisfies $λ (G) \leq \sqrt{m}$ , equality holds if and only if G is a complete bipartite graph. Lin, Ning and Wu (2021) extended this result as: If G is a $C_{3}$ -free non-bipartite graph of size m, then $λ (G) \leq \sqrt{m - 1}$ , equality holds if and only if $G ≅ C_{5}$ . This result was extended by Li, Peng (2022) and Sun, Li (2023), independently, as the following: If G is a ${C_{3}, C_{5}}$ -free non-bipartite graph with m edges, then $λ (G) \leq λ (S_{3} (K_{2, \frac{m - 3}{2}}))$ , equality holds if and only if m is odd and $G ≅ S_{3} (K_{2, \frac{m - 3}{2}})$ , where $S_{3} (K_{2, \frac{m - 3}{2}})$ is obtained from $K_{2, \frac{m - 3}{2}}$ by replacing one of its edges by a path of length 4. This upper bound could be attained only if m is odd, since the extremal graph $S_{3} (K_{2, \frac{m - 3}{2}})$ is well-defined only in this case. Thus, it is interesting to determine the spectral extremal graph when m is even. Sun and Li (2023) proposed the following question: Determine the graphs attaining the maximum spectral radius o

Nosal (1970) 和 Nikiforov (2002) 发现，如果图 G 是大小为 m 的无 C3 图，则 G 的谱半径满足 λ(G)≤m，且只有当且仅当 G 是一个完整的双向图时，等式才成立。Lin、Ning 和 Wu (2021) 将这一结果扩展为：如果 G 是大小为 m 的无 C3 非双向图，那么只有当 G≅C5 时，λ(G)≤m-1，等式成立。李鹏（2022）和孙莉（2023）分别将这一结果扩展如下：如果 G 是一个有 m 条边的无{C3,C5}非双面图，那么λ(G)≤λ(S3(K2,m-32))，当且仅当 m 为奇数且 G≅S3(K2,m-32)，其中 S3(K2,m-32) 是通过将 K2,m-32 的一条边替换为长度为 4 的路径得到的。只有当 m 为奇数时才能达到这个上限，因为极值图 S3(K2,m-32) 只有在这种情况下才定义明确。因此，确定 m 为偶数时的谱极值图是很有意义的。孙和李（2023 年）提出了以下问题：在本论文中，我们将回答 m≥150 时的这个问题。我们的证明技术主要基于图的特征值的 Cauchy 交错定理，并借助于宁和翟在特征值和图的大小方面的三角形计数法，以及 Lou, Lu 和 Huang (2023) 的特征向量法。

引用次数: 0

On Rado numbers for equations with unit fractions 关于有单位分数的方程的拉多数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-07-09 DOI: 10.1016/j.disc.2024.114156

Collier Gaiser

Let $f_{r} (k)$ be the smallest positive integer n such that every r-coloring of ${1, 2, \dots, n}$ has a monochromatic solution to the nonlinear equation $1 / x_{1} + \dots + 1 / x_{k} = 1 / y,$ where $x_{1}, \dots, x_{k}$ are not necessarily distinct. Brown and Rödl [3] proved that $f_{2} (k) = O (k^{6})$ . In this paper, we prove that $f_{2} (k) = O (k^{3})$ . The main ingredient in our proof is a finite set $A \subseteq N$ such that every 2-coloring of A has a monochromatic solution to the linear equation $x_{1} + \dots + x_{k} = y$ and the least common multiple of A is sufficiently small. This approach can also be used to study $f_{r} (k)$ with $r > 2$ . For example, a recent result of Boza et al. [2] implies that $f_{3} (k) = O (k^{43})$ .

设 fr(k) 是最小的正整数 n，使得{1,2,...,n}的每一个 r 色都有非线性方程 1/x1+⋯+1/xk=1/y 的单色解，其中 x1,...xk 不一定是不同的。Brown 和 Rödl [3] 证明了 f2(k)=O(k6) 。本文将证明 f2(k)=O(k3)。我们证明的主要要素是一个有限集 A⊆N，使得 A 的每个 2 色都有线性方程 x1+⋯+xk=y 的单色解，并且 A 的最小公倍数足够小。例如，Boza 等人[2]的最新结果暗示 f3(k)=O(k43)。

引用次数: 0

On two conjectures about the intersection of longest paths and cycles 关于最长路径与循环相交的两个猜想

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-07-03 DOI: 10.1016/j.disc.2024.114148

Juan Gutiérrez , Christian Valqui

A conjecture attributed to Smith states that every two longest cycles in a k-connected graph intersect in at least k vertices. In this paper, we show that every two longest cycles in a k-connected graph on n vertices intersect in at least $\min {n, 8 k - n - 16}$ vertices, which confirms Smith's conjecture when $k \geq (n + 16) / 7$ . An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either $k \leq 7$ or $k \geq (n + 9) / 7$ .

Smith 提出的一个猜想是：在 k 个连通图中，每两个最长循环至少在 k 个顶点上相交。在本文中，我们证明了当 k≥(n+16)/7 时，n 个顶点上 k 个连接图中的每两个最长循环至少相交于 min{n,8k-n-16} 个顶点，这证实了 Smith 的猜想。希普钦（Hippchen）针对路径而非循环提出了类似猜想。通过简单的还原，我们将这两个猜想联系起来，证明当 k≤7 或 k≥(n+9)/7 时，希普钦的猜想是有效的。

引用次数: 0

Generalized Turán results for edge blow-up of star forests 星形森林边缘膨胀的广义图兰结果

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-07-02 DOI: 10.1016/j.disc.2024.114149

Yuanpei Wang , Zhenyu Ni , Yichong Liu , Liying Kang

Given a graph H and an integer $p \geq 2$ , the edge blow-up $H^{p + 1}$ of H is the graph obtained by replacing each edge in H with a clique of order $p + 1$ , where the new vertices of the cliques are all distinct. The generalized Turán number $e x (n, K_{m}, F)$ denote the maximum number of copies of $K_{m}$ in an n-vertex F-free graph. In this paper, we determine the exact value of generalized Turán number for edge blow-up of star forests and characterize the unique graph for sufficiently large n.

给定一个图 H 和一个整数 p≥2，H 的边炸开 Hp+1 是将 H 中的每条边替换为 p+1 阶的簇后得到的图，其中簇的新顶点都是不同的。广义图兰数 ex(n,Km,F) 表示无 n 个顶点的 F 图中 Km 的最大副本数。本文确定了星形森林边缘膨胀的广义图兰数的精确值，并描述了足够大 n 的唯一图的特征。

引用次数: 0

A construction of directed strongly regular graphs with parameters (63,11,8,1,2) 构建参数为 (63,11,8,1,2) 的有向强规则图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-06-28 DOI: 10.1016/j.disc.2024.114146

Andries E. Brouwer , Dean Crnković , Andrea Švob

In this paper, we prove the existence of directed strongly regular graphs with parameters $(63, 11, 8, 1, 2)$ . We construct a pair of nonisomorphic dsrg(63,11,8,1,2), where one is obtained from the other by reversing all arrows. Both directed strongly regular graphs have $L_{2} (8) : 3$ as the full automorphism group.

本文证明了参数为 (63,11,8,1,2) 的有向强规则图的存在性。我们构造了一对非同构的 dsrg(63,11,8,1,2)，其中一个是通过反转所有箭头从另一个得到的。这两个有向强规则图的全自形群都是 L2(8):3。

引用次数: 0

A network flow approach to a common generalization of Clar and Fries numbers 克拉克数和弗里斯数通用广义化的网络流方法

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-06-26 DOI: 10.1016/j.disc.2024.114145

Erika Bérczi-Kovács , András Frank

Clar number and Fries number are two thoroughly investigated parameters of plane graphs emerging from mathematical chemistry to measure stability of organic molecules. First, we introduce a common generalization of these two concepts for bipartite plane graphs, and then we extend it further to the notion of source-sink pairs of subsets of nodes in a general (not necessarily planar) directed graph. The main result is a min-max formula for the maximum weight of a source-sink pair. The proof is based on the recognition that the convex hull of source-sink pairs can be obtained as the projection of a network tension polyhedron. The construction makes it possible to apply any standard cheapest network flow algorithm to compute both a maximum weight source-sink pair and a minimizer of the dual optimization problem formulated in the min-max theorem. As a consequence, our approach gives rise to the first purely combinatorial, strongly polynomial algorithm to compute a largest (or even a maximum weight) Fries-set of a perfectly matchable plane bipartite graph and an optimal solution to the dual minimization problem.

克拉数和弗里斯数是数学化学中出现的平面图的两个经过深入研究的参数，用于测量有机分子的稳定性。首先，我们介绍了这两个概念在二方平面图中的通用概括，然后进一步扩展到一般（不一定是平面）有向图中节点子集的源-汇对概念。主要结果是源-汇对最大权重的最小-最大公式。证明的基础是认识到源-汇对的凸壳可以作为网络张力多面体的投影来获得。这种构造使得应用任何标准的最廉价网络流算法来计算最大权重源-汇对和最小最大定理中提出的对偶优化问题的最小值成为可能。因此，我们的方法产生了第一种纯组合、强多项式算法，可用于计算完全匹配平面双啮合图的最大（甚至最大权重）弗里斯集以及对偶最小化问题的最优解。

引用次数: 0

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