Discrete Mathematics最新文献

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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

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

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

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

<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)。

引用次数: 0

Disconnected forbidden pairs force supereulerian graphs to be hamiltonian 断开的禁止对迫使超立体图成为哈密顿图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114301

A graph is said to be supereulerian if it has a spanning eulerian subgraph, i.e., a spanning connected even subgraph. A graph is called hamiltonian if it contains a spanning cycle. A graph is said to be

{R, S}

-free if it does not contain R or S as an induced subgraph. Yang et al. characterized all pairs of connected graphs

R, S

such that every supereulerian

{R, S}

-free graph is hamiltonian. In this paper, we consider disconnected forbidden graph

R, S

. We characterize all pairs of disconnected graphs

R, S

such that every supereulerian

{R, S}

-free graph of sufficiently large order is hamiltonian. Applying this result, we also characterize all forbidden pairs for the existence of a Hamiltonian cycle in 2-edge connected graphs.

如果一个图有一个跨越的优勒子图，即一个跨越的连通偶数子图，则称该图为超优勒图。如果一个图包含一个跨循环，则称为哈密顿图。如果一个图不包含 R 或 S 作为诱导子图，则称其为无{R,S}图。Yang等人描述了所有连通图R,S的特征，即每个无超循环{R,S}图都是哈密顿图。在本文中，我们考虑断开的禁止图 R,S。我们描述了所有成对的断开图 R,S 的特征，即每个阶数足够大的无超线性 {R,S} 图都是哈密顿图。应用这一结果，我们还描述了在 2 边相连图中存在哈密顿循环的所有禁止图对。

引用次数: 0

Rigid frameworks with dilation constraints 具有扩张约束的刚性框架

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-31 DOI: 10.1016/j.disc.2024.114304

We consider the rigidity and global rigidity of bar-joint frameworks in Euclidean d-space under additional dilation constraints in specified coordinate directions. In this setting we obtain a complete characterisation of generic rigidity. We then consider generic global rigidity. In particular, we provide an algebraic sufficient condition and a weak necessary condition. We also construct a large family of globally rigid frameworks and conjecture a combinatorial characterisation when most coordinate directions have dilation constraints.

我们考虑了欧几里得 d 空间中棒关节框架在指定坐标方向的额外扩张约束下的刚度和全局刚度。在这种情况下，我们得到了一般刚性的完整特征。然后，我们考虑一般全局刚度。特别是，我们提供了一个代数充分条件和一个弱必要条件。我们还构建了一个庞大的全局刚性框架族，并猜想了当大多数坐标方向都有扩张约束时的组合特征。

引用次数: 0

A note on locating-dominating sets in twin-free graphs 关于无孪生图中定位支配集的说明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.disc.2024.114297

In this short note, we prove that every twin-free graph on n vertices contains a locating-dominating set of size at most

⌈ \frac{5}{8} n ⌉

. This improves the earlier bound of

⌊ \frac{2}{3} n ⌋

due to Foucaud, Henning, Löwenstein and Sasse from 2016, and makes some progress towards the well-studied locating-dominating conjecture of Garijo, González and Márquez.

在这篇短文中，我们证明了 n 个顶点上的每个无孪生图都包含一个大小至多为⌈58n⌉的定位支配集。这改进了 Foucaud、Henning、Löwenstein 和 Sasse 在 2016 年提出的⌊23n⌋ 的早期约束，并在实现 Garijo、González 和 Márquez 的定位支配猜想方面取得了一些进展。

引用次数: 0

Generation of 3-connected, planar line graphs 生成三连平面线图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-30 DOI: 10.1016/j.disc.2024.114302

We classify and construct all line graphs that are 3-polytopes (planar and 3-connected). Apart from a few special cases, they are all obtained starting from the medial graphs of cubic (i.e., 3-regular) 3-polytopes, by applying two types of graph transformations. This is similar to the generation of other subclasses of 3-polytopes [6], [13].

我们对所有属于 3 多面体（平面和 3 连接）的线图进行了分类和构造。除少数特例外，它们都是从立方（即 3-不规则）3-多面体的中间图开始，通过应用两种图形变换得到的。这与生成其他子类的 3 多面体[6], [13]类似。

引用次数: 0

Light 3-faces in 3-polytopes without adjacent triangles 无相邻三角形的 3 多面体中的光 3 面

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.disc.2024.114299

Over the last decades, a lot of research has been devoted to structural and coloring problems on plane graphs that are sparse in this or that sense.

In this note we deal with the densest among sparse 3-polytopes, namely those having no adjacent 3-cycles. Borodin (1996) proved that such 3-polytopes have a vertex of degree at most 4 and, moreover, an edge with the degree-sum of its end-vertices at most 9, where both bounds are sharp.

d (v)

denote the degree of a vertex v. An edge

e = x y

in a 3-polytope is an

(i, j)

-edge if

d (x) \leq i

and

d (y) \leq j

. The well-known (3,5;4,4)-Archimedean solid corresponds to a plane quadrangulation in which every edge joins a 3-vertex with a 5-vertex.

We prove that every 3-polytope with neither adjacent 3-cycles nor

(3, 5)

-edges has a 3-face with the degree-sum of its incident vertices (weight) at most 16, which bound is sharp.

在过去的几十年里，很多人致力于研究在这种或那种意义上稀疏的平面图的结构和着色问题。在本论文中，我们将讨论稀疏 3 多面体中最密集的一种，即没有相邻 3 循环的多面体。鲍罗丁（Borodin，1996 年）证明了这种 3 多面体的顶点阶数最多为 4，而且边的末端顶点阶数总和最多为 9，这两个界限都很尖锐。我们证明了每一个既没有相邻 3 循环也没有 (3,5) 边的 3 多面体都有一个其入射顶点度数和（权重）最多为 16 的 3 面，这个约束是尖锐的。

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引用次数: 0

Counting spanning trees of multiple complete split-like graph containing a given spanning forest 对包含给定生成林的多个完整分裂样图的生成树进行计数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-24 DOI: 10.1016/j.disc.2024.114300

The multiple complete split-like graph

M C S_{b, s}^{a}

is the join of an empty graph

{\overline{K}}_{a}

and s copies of complete graph

K_{b}

. In this article, we obtain the formulas for the number of spanning trees of

M C S_{b, s}^{a}

containing a given spanning forest when

s = 1

and 2. Particularly, when

s = 1

, our result derives the number of spanning trees of complete split graph containing a given spanning forest, thereby extending Moon's result [19].

多重完整分裂样图 MCSb,sa 是空图 K‾a 和完整图 Kb 的 s 个副本的连接。在本文中，我们得到了当 s=1 和 2 时，MCSb,sa 中包含给定生成林的生成树的数量公式。特别是当 s=1 时，我们的结果推导出了包含给定生成林的完整分裂图的生成树数，从而扩展了 Moon 的结果 [19]。

引用次数: 0

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Discrete Mathematics

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