Discrete Mathematics最新文献

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Six-flows of signed graphs with frustration index three 挫折指数为 3 的有符号图形的六次流动

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-12 DOI: 10.1016/j.disc.2024.114325

You Lu , Rong Luo , Cun-Quan Zhang

Bouchet's 6-flow conjecture states that every flow-admissible signed graph admits a nowhere-zero 6-flow. Seymour's 6-flow theorem states that the conjecture holds for balanced signed graphs. Rollová et al. show that every flow-admissible signed graph with frustration index two admits a nowhere-zero 7-flow, where the frustration index of a signed graph is the smallest number of edges whose deletion leaves a balanced signed graph. Wang et al. improve the result to 6-flows. In this paper, we further extend these results, and confirm Bouchet's 6-flow conjecture for signed graphs with frustration index three. There are infinitely many signed graphs with frustration index three admitting a nowhere-zero 6-flow but no 5-flow. From the point of view of flow theory, signed graphs with frustration index two are very similar to those of ordinary graphs. However, there are significant differences between ordinary graphs and signed graphs with frustration index three.

Bouchet 的 6 流猜想指出，每个流容许的有符号图都有一个无处为零的 6 流。西摩的 6 流定理指出，该猜想对于平衡有符号图成立。Rollová 等人的研究表明，每一个沮度指数为 2 的流动可容许有符号图都有一个无处为零的 7 流，其中有符号图的沮度指数是删除后留下平衡有符号图的边的最小数目。Wang 等人将这一结果改进为 6 流。在本文中，我们进一步扩展了这些结果，并证实了 Bouchet 对沮度指数为 3 的有符号图的 6 流猜想。有无穷多个沮度指数为三的有符号图允许无处为零的 6 流，但不允许 5 流。从流论的角度来看，挫折指数为 2 的有符号图与普通图非常相似。然而，普通图与挫折指数为三的有符号图之间存在显著差异。

引用次数: 0

Partial packing coloring and quasi-packing coloring of the triangular grid 三角形网格的部分堆积着色和准堆积着色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-12 DOI: 10.1016/j.disc.2024.114308

Hubert Grochowski, Konstanty Junosza-Szaniawski

The concept of packing coloring in graph theory is motivated by the challenge of frequency assignment in radio networks. This approach entails assigning positive integers to vertices, with the requirement that for any given label (color) i, the distance between any two vertices sharing this label must exceed i. Recently, after over 20 years of intensive research, the minimal number of colors needed for packing coloring of an infinite square grid has been established to be 15. Moreover, it is known that a hexagonal grid requires a minimum of 7 colors for packing coloring, and a triangular grid is not colorable with any finite number of colors in a packing way.

Therefore, two questions come to mind: What fraction of a triangular grid can be colored in a packing model, and how much do we need to weaken the condition of packing coloring to enable coloring a triangular grid with a finite number of colors?

With the partial help of the Mixed Integer Linear Programming (MILP) solver, we have proven that it is possible to color at least 72.8% but no more than 82.2% of a triangular grid in a packing way.

Additionally, we have investigated the relaxation of packing coloring, called quasi-packing coloring, which is a special case of S-packing coloring. We have established that the S-packing chromatic number for the triangular grid, where

S = (1, 1, 2, 3, . . .)

, is between 11 and 33. Furthermore, we have proven that the aforementioned sequence S is the best possible in some sense.

We have also considered the partial packing and quasi-packing coloring of an infinite hypercube and present several open problems for other classes of graphs.

图论中的打包着色概念源于无线电网络中频率分配所面临的挑战。这种方法需要为顶点分配正整数，并要求对于任何给定的标签（颜色）i，共享该标签的任何两个顶点之间的距离必须超过 i。最近，经过 20 多年的深入研究，确定了无限正方形网格打包着色所需的最少颜色数为 15 种。此外，已知六边形网格打包着色至少需要 7 种颜色，而三角形网格无法用任何有限数量的颜色打包着色：因此，我们想到了两个问题：在打包模型中，有多大比例的三角形网格可以着色；我们需要在多大程度上削弱打包着色的条件，才能用有限数量的颜色给三角形网格着色？此外，我们还研究了打包着色的松弛，即准打包着色，它是 S-打包着色的一个特例。我们确定了三角形网格（其中 S=（1,1,2,3,......）的 S-堆积色度数介于 11 和 33 之间。此外，我们还证明了上述序列 S 在某种意义上是最好的。我们还考虑了无限超立方体的部分堆积和准堆积着色，并提出了其他类别图的几个未决问题。

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引用次数: 0

Generalized Mneimneh sums and their application to multiple polylogarithms 广义姆奈奈和及其在多重多项式中的应用

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-08 DOI: 10.1016/j.disc.2024.114318

Marian Genčev

The purpose of this paper is the study of the binomial sum

\sum_{k = 1}^{n} (\begin{matrix} n \\ k \end{matrix}) \cdot H_{k}^{(s)} (a) \cdot p^{k} \cdot {(1 - p)}^{n - k},

where

H_{k}^{(s)} (a) = \sum_{i = 1}^{k} a^{i} / i^{s}

denotes the parameterized analogue of the k-th harmonic number of order s. For

a = s = 1

, these binomial sums were investigated by Mneimneh, who gave a probabilistic interpretation related to hiring problems. We present a generalization of Mneimneh's summation formula and establish several new identities and a connection of these sums with specific multiple polylogarithms, called unit Euler sums, based upon the Toeplitz limit theorem.

本文的目的是研究二项式和∑k=1n(nk)⋅Hk(s)(a)⋅pk⋅(1-p)n-k，其中 Hk(s)(a)=∑i=1kai/is 表示阶数 s 的 k 次谐波数的参数化类似值。对于 a=s=1，这些二项式和由 Mneimneh 研究，他给出了与招聘问题有关的概率解释。我们提出了 Mneimneh 求和公式的广义化，并根据托普利兹极限定理，建立了这些和与特定多重多项式（称为单位欧拉和）之间的若干新特性和联系。

引用次数: 0

On combinatorial and hypergeometric approaches toward second-order difference equations 关于二阶差分方程的组合和超几何方法

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-08 DOI: 10.1016/j.disc.2024.114316

John M. Campbell

Laohakosol et al. recently introduced enumerative techniques based on second-order difference equations to prove a number of conjectured evaluations for polynomial continued fractions generated by the Ramanujan Machine. Each of the discrete difference equations required according to the combinatorial approach employed by Laohakosol et al. can be solved in an explicit way according to an alternative and hypergeometric-based approach that we apply to prove further conjectures produced by the Ramanujan Machine. An advantage of our hypergeometric approach, compared to the methods of Laohakosol et al. and compared to solving for ODEs satisfied by formal power series corresponding to the Euler–Wallis recursions, is given by the explicit evaluations for the nonlinear difference equations that we obtain.

最近，Laohakosol 等人引入了基于二阶差分方程的枚举技术，证明了由 Ramanujan 机生成的多项式连续分数的一系列猜想评估。根据 Laohakosol 等人采用的组合方法，所需的每个离散差分方程都可以根据另一种基于超几何的方法显式求解。与 Laohakosol 等人的方法相比，以及与通过与欧拉-瓦利斯递推对应的形式幂级数求解 ODEs 相比，我们的超几何方法的优势在于我们得到的非线性差分方程的显式求值。

引用次数: 0

Packing directed cycles of specified odd length into digraphs and alternating cycles into bipartite graphs 将指定奇数长度的有向循环打包到数图中，将交替循环打包到二方图中

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-06 DOI: 10.1016/j.disc.2024.114306

Shuya Chiba , Koshin Yoshida

In this paper, we prove the following result. For given integers

k \geq 1, t \geq 0

with

k \geq t

and an odd integer

ℓ \geq 3

, there exists an integer

n_{0} = n_{0} (k, t, ℓ)

satisfying the following: If D is a digraph of order

n \geq n_{0}

, and if

d_{D}^{+} (u) + d_{D}^{-} (v) \geq n + t

for every two distinct vertices u and v with

(u, v) \notin A (D)

, then D contains k vertex-disjoint directed cycles of length ℓ or

ℓ + 1

such that at least t of them are of length ℓ. This is a common extension of the results obtained by Brandt et al. (1997) and, Chiba and Yamashita (2018). We also discuss the relation between our result and problems on packing alternating cycles into bipartite graphs.

本文将证明以下结果。对于给定整数 k≥1,t≥0，且 k≥t 和奇整数 ℓ≥3，存在满足以下条件的整数 n0=n0(k,t,ℓ)：如果 D 是阶数 n≥n0 的数图，且对于每两个不同的顶点 u 和 v，且 (u,v)∉A(D) 的 dD+(u)+dD-(v)≥n+t ，那么 D 包含长度为 ℓ 或 ℓ+1 的 k 个顶点相交的有向循环，且其中至少有 t 个循环的长度为 ℓ。这是对 Brandt 等人（1997）以及 Chiba 和 Yamashita（2018）所获结果的常见扩展。我们还讨论了我们的结果与将交替循环打包进双方形图问题之间的关系。

引用次数: 0

Refinements of degree conditions for the existence of a spanning tree without small degree stems 细化无小度干生成树存在的度数条件

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-11-05 DOI: 10.1016/j.disc.2024.114307

Michitaka Furuya , Akira Saito , Shoichi Tsuchiya

A spanning tree of a graph without vertices of degree 2 is called a homeomorphically irreducible spanning tree (or a HIST) of the graph. Albertson et al. (1990) [1] gave a minimum degree condition for the existence of a HIST, and recently, Ito and Tsuchiya (2022) [11] found a sharp degree-sum condition for the existence of a HIST. In this paper, we refine these results, and extend the first one to a spanning tree in which no vertex other than the endvertices has small degree.

没有阶数为 2 的顶点的图的生成树称为图的同构不可还原生成树（或 HIST）。Albertson 等人（1990）[1] 给出了 HIST 存在的最小度条件，最近，Ito 和 Tsuchiya（2022）[11] 发现了 HIST 存在的尖锐度和条件。在本文中，我们完善了这些结果，并将第一个结果扩展到了除端顶点外没有其他顶点具有小度的生成树。

引用次数: 0

Every 3-connected {K1,3,Γ3}-free graph is Hamilton-connected 每个无 3 连接的 {K1,3,Γ3} 图都是汉密尔顿连接的

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114305

Adam Kabela, Zdeněk Ryjáček, Mária Skyvová, Petr Vrána

We show that every 3-connected

{K_{1, 3}, Γ_{3}}

-free graph is Hamilton-connected, where

Γ_{3}

is the graph obtained by joining two vertex-disjoint triangles with a path of length 3. This resolves one of the two last open cases in the characterization of pairs of connected forbidden subgraphs implying Hamilton-connectedness. The proof is based on a new closure technique, developed in a previous paper, and on a structural analysis of small subgraphs, cycles and paths in line graphs of multigraphs. The most technical steps of the analysis are computer-assisted.

我们证明了每一个无 3 连接的 {K1,3,Γ3} 图都是汉密尔顿连接的，其中Γ3 是用长度为 3 的路径连接两个顶点相交的三角形所得到的图。这解决了暗示汉密尔顿连接性的成对连接的禁止子图的特征描述中最后两个开放案例之一。证明基于前一篇论文中开发的一种新的闭合技术，以及对多图行图中的小子图、循环和路径的结构分析。分析的大部分技术步骤都由计算机辅助完成。

引用次数: 0

Completely regular codes with covering radius 1 and the second eigenvalue in 3-dimensional Hamming graphs 覆盖半径为 1 的完全正则码和三维汉明图中的第二特征值

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114296

Ivan Mogilnykh, Anna Taranenko, Konstantin Vorob'ev

We obtain a classification of completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs

H (3, q)

up to q and intersection array. Due to the works of Meyerowitz, Mogilnykh, and Valyuzenich, our result completes the classifications of completely regular codes with covering radius 1 and the second eigenvalue in the Hamming graphs

H (n, q)

for any n and completely regular codes with covering radius 1 in

H (3, q)

我们获得了覆盖半径为 1 且在汉明图 H(3,q) 中具有第二特征值（直到 q 和交集阵列）的完全正则码的分类。由于 Meyerowitz、Mogilnykh 和 Valyuzenich 的工作，我们的结果完成了对任意 n 的汉明图 H(n,q) 中覆盖半径为 1 和第二特征值的完全正则码以及 H(3,q) 中覆盖半径为 1 的完全正则码的分类。

引用次数: 0

On finding the largest minimum distance of locally recoverable codes: A graph theory approach 关于寻找局部可恢复编码的最大最小距离：图论方法

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114298

Majid Khabbazian , Muriel Médard

The

[n, k, r]

-Locally recoverable codes (LRC) studied in this work are a well-studied family of

[n, k]

linear codes for which the value of each symbol can be recovered by a linear combination of at most r other symbols. In this paper, we study the LMD problem, which is to find the largest possible minimum distance of

[n, k, r]

-LRCs, denoted by

D (n, k, r)

. LMD can be approximated within an additive term of one—it is known that

D (n, k, r)

is equal to either

d^{⁎}

d^{⁎} - 1

, where

d^{⁎} = n - k - ⌈ \frac{k}{r} ⌉ + 2

. Moreover, for a range of parameters, it is known precisely whether the distance

D (n, k, r)

d^{⁎}

d^{⁎} - 1

. However, the problem is still open despite a significant effort. In this work, we convert LMD to an equivalent simply-stated problem in graph theory. Using this conversion, we show that an instance of LMD is at least as hard as computing the size of a maximal graph of high girth, a hard problem in extremal graph theory. This is an evidence that LMD—although can be approximated within an additive term of one—is hard to solve in general. As a positive result, thanks to the conversion and the exiting results in extremal graph theory, we solve LMD for a range of code parameters that has not been solved before.

本文所研究的 [n,k,r]-Locally recoverable 编码（LRC）是一个经过深入研究的 [n,k] 线性编码系列，其中每个符号的值最多可以通过 r 个其他符号的线性组合来恢复。本文研究的是 LMD 问题，即找出 [n,k,r]-LRC 的最大可能最小距离，用 D(n,k,r) 表示。众所周知，D(n,k,r) 等于 d⁎ 或 d⁎-1，其中 d⁎=n-k-⌈kr⌉+2。此外，对于一系列参数来说，距离 D(n,k,r) 是 d⁎ 还是 d⁎-1，都是已知的。然而，尽管做了大量工作，这个问题仍然没有解决。在这项工作中，我们将 LMD 转换为图论中的等价简单问题。利用这种转换，我们证明了 LMD 实例的难度至少与计算高周长最大图的大小相当，而计算高周长最大图的大小是极值图论中的一个难题。这证明了 LMD 虽然可以在 1 的加法项内近似，但一般来说很难求解。作为一个积极的结果，得益于极值图论中的转换和现有成果，我们解决了一系列代码参数下的 LMD 问题，而这在以前是没有解决过的。

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引用次数: 0

Extremal bounds for pattern avoidance in multidimensional 0-1 matrices 多维 0-1 矩阵中模式规避的极值界限

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114303

Jesse Geneson , Shen-Fu Tsai

<div><div>A 0-1 matrix M contains another 0-1 matrix P if some submatrix of M can be turned into P by changing any number of 1-entries to 0-entries. The 0-1 matrix M is <math><mi>P</mi></math>-saturated where <math><mi>P</mi></math> is a family of 0-1 matrices if M avoids every element of <math><mi>P</mi></math> and changing any 0-entry of M to a 1-entry introduces a copy of some element of <math><mi>P</mi></math>. The extremal function <math><mi>ex</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math> and saturation function <math><mi>sat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math> are the maximum and minimum possible number of 1-entries in an <math><mi>n</mi><mo>×</mo><mi>n</mi></math> <math><mi>P</mi></math>-saturated 0-1 matrix, respectively, and the semisaturation function <math><mi>ssat</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>P</mi><mo>)</mo></math> is the minimum possible number of 1-entries in an <math><mi>n</mi><mo>×</mo><mi>n</mi></math> <math><mi>P</mi></math>-semisaturated 0-1 matrix M, i.e., changing any 0-entry in M to a 1-entry introduces a new copy of some element of <math><mi>P</mi></math>.</div><div>We study these functions of multidimensional 0-1 matrices. In particular, we give upper bounds on parameters of minimally non-<math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math> d-dimensional 0-1 matrices, generalized from minimally nonlinear 0-1 matrices in two dimensions, and we show the existence of infinitely many minimally non-<math><mi>O</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>)</mo></math> d-dimensional 0-1 matrices with all dimensions of length greater than 1. For any positive integers <math><mi>k</mi><mo>,</mo><mi>d</mi></math> and integer <math><mi>r</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mi>d</mi><mo>−</mo><mn>1</mn><mo>]</mo></math>, we construct a family of d-dimensional 0-1 matrices with both extremal function and saturation function exactly <math><mi>k</mi><msup><mrow><mi>n</mi></mrow><mrow><mi>r</mi></mrow></msup></math> for sufficiently large n. We show that no family of d-dimensional 0-1 matrices has saturation function strictly between <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> and <math><mi>Θ</mi><mo>(</mo><mi>n</mi><mo>)</mo></math> and we construct a family of d-dimensional 0-1 matrices with bounded saturation function and extremal function <math><mi>Ω</mi><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>d</mi><mo>−</mo><mi>ϵ</mi></mrow></msup><mo>)</mo></math></spa

如果 M 的某个子矩阵可以通过将任意数量的 1 条目变为 0 条目而变成 P，则 0-1 矩阵 M 包含另一个 0-1 矩阵 P。如果 M 避开了 P 的每个元素，并且将 M 的任意 0 条目改为 1 条目都会引入 P 的某个元素的副本，那么 0-1 矩阵 M 就是 P 饱和的，其中 P 是 0-1 矩阵族。极值函数 ex(n,P) 和饱和函数 sat(n,P) 分别是 n×n P 饱和 0-1 矩阵中 1 条目的最大可能数目和最小可能数目，而半饱和函数 ssat(n,P) 是 n×n P 半饱和 0-1 矩阵 M 中 1 条目的最小可能数目，即、我们研究多维 0-1 矩阵的这些函数。特别是，我们给出了最小非 O（nd-1）d 维 0-1 矩阵参数的上限，这是从二维最小非线性 0-1 矩阵推广而来的；我们还证明了存在无限多的最小非 O（nd-1）d 维 0-1 矩阵，且所有维的长度都大于 1。对于任意正整数 k,d 和整数 r∈[0,d-1]，我们构造了一个 d 维 0-1 矩阵族，其极值函数和饱和函数在足够大的 n 条件下正好为 knr。我们证明没有一个 d 维 0-1 矩阵族的饱和函数严格介于 O(1) 和 Θ(n) 之间，并且我们构造了一个 d 维 0-1 矩阵族，其饱和函数和极值函数 Ω(nd-ϵ) 对于任意 ϵ>0 都是有界的。对于某个整数 r∈[0,d-1]，我们证明其半饱和函数总是 Θ(nr)。

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全部化学•材料生命科学医学物理工程技术环境•农林材料科学地球科学法学管理学化学环境科学与生态学计算机科学教育学经济学农林科学人文科学生物学数学物理与天体物理心理学综合性期刊其他工业工程理学历史学农学文学信息工程

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