Discrete Mathematics最新文献

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Hamiltonian decompositions of the wreath product of hamiltonian decomposable digraphs 哈密顿可分解有向图环积的哈密顿分解

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-27 DOI: 10.1016/j.disc.2026.115012

Alice Lacaze-Masmonteil

We confirm most open cases of a conjecture that first appeared in Alspach et al. (1987) which stipulates that the wreath (lexicographic) product of two hamiltonian decomposable directed graphs is also hamiltonian decomposable. Specifically, we show that the wreath product of a hamiltonian decomposable directed graph G, such that

| V (G) |

is even and

| V (G) | ⩾ 2

, with a hamiltonian decomposable directed graph H, such that

| V (H) | ⩾ 4

, is also hamiltonian decomposable except possibly when G is a directed cycle and H is a directed graph of odd order that admits a decomposition into c directed hamiltonian cycle where c is odd and

3 ⩽ c ⩽ | V (H) | - 2

我们证实了Alspach等人（1987）首次提出的一个猜想的大多数开放情况，该猜想规定两个哈密顿可分解有向图的环（字典）积也是哈密顿可分解的。具体地说，我们表明哈密顿可分解有向图G的环积，使得|V(G)|是偶数，|V(G)|小于2，具有哈密顿可分解有向图H，使得|V(H)|小于4，也是哈密顿可分解的，除非可能当G是有向循环并且H是奇阶有向图允许分解成c有向哈密顿循环其中c是奇数并且3≤c≤|V(H)|−2。

引用次数: 0

The Turán number of Berge-linear forests in hypergraphs 超图中berge -线性森林的Turán个数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-05 DOI: 10.1016/j.disc.2025.114971

Lin-Peng Zhang , Hajo Broersma , Ligong Wang

<div><div>Let <math><mi>F</mi></math> be a family of graphs, and let H be a hypergraph. H is called a Berge-<math><mi>F</mi></math> if for some <math><mi>F</mi><mo>∈</mo><mi>F</mi></math>, there exists an injection <math><mi>θ</mi><mo>:</mo><mi>V</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>→</mo><mi>V</mi><mo>(</mo><mi>H</mi><mo>)</mo></math> and a bijection <math><mi>ϕ</mi><mo>:</mo><mi>E</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>→</mo><mi>E</mi><mo>(</mo><mi>H</mi><mo>)</mo></math> such that <math><mo>{</mo><mi>θ</mi><mo>(</mo><mi>u</mi><mo>)</mo><mo>,</mo><mi>θ</mi><mo>(</mo><mi>v</mi><mo>)</mo><mo>}</mo><mo>⊆</mo><mi>ϕ</mi><mo>(</mo><mi>e</mi><mo>)</mo></math> for each <math><mi>e</mi><mo>=</mo><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo><mo>∈</mo><mi>E</mi><mo>(</mo><mi>F</mi><mo>)</mo></math>. H is called Berge-<math><mi>F</mi></math>-free if H contains no subhypergraph isomorphic to any Berge-<math><mi>F</mi></math>. The Turán number of a Berge-<math><mi>F</mi></math>, denoted by <math><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mtext>Berge-</mtext><mi>F</mi><mo>)</mo></math>, is defined as the maximum number of edges in an n-vertex Berge-<math><mi>F</mi></math>-free r-uniform hypergraph. A linear forest is a graph all components of which are paths or isolated vertices. Denote by <math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math> the family of all linear forests containing n vertices and k edges. In this paper, we determine the value of <math><msub><mrow><mi>ex</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mtext>Berge-</mtext><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub><mo>)</mo></math> for the cases <math><mn>3</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mo>⌈</mo><mfrac><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo><mo>−</mo><mn>2</mn></math> and <math><mi>r</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn>1</mn></math>. Furthermore, we characterize the extremal hypergraphs for the cases <math><mn>3</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mfrac><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mn>3</mn></math> and <math><mi>r</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn>1</mn></math>, when k is odd, and for the cases <math><mn>3</mn><mo>≤</mo><mi>r</mi><mo>≤</mo><mfrac><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mn>2</mn></math> and <math><mi>r</mi><mo>≥</mo><mi>k</mi><mo>+</mo><mn>1</mn></math>, when k is even. We establish an upper bound on <math><msub><mrow><mi>ex</mi></mrow><mrow><

设F是一个图族，设H是一个超图。如果对于某些F∈F，存在一个注入θ:V(F)→V(H)和一个双注入φ:E(F)→E(H)，使得对于每个E ={u, V}∈E(F), {θ(u),θ(V)}≥≥φ (E)， H称为Berge-F。如果H不包含与任何Berge-F同构的子超图，则称H为无Berge-F。Berge-F的Turán个数，用exr（n,Berge-F）表示，定义为n顶点无Berge-F的r-均匀超图的最大边数。线性森林是一个图，它的所有组成部分都是路径或孤立的顶点。用Ln k表示所有包含n个顶点和k条边的线性森林族。在本文中,我们确定的价值exr (Berge-Ln n, k)的情况下3 r≤≤⌈k + 12⌉−2 r≥k + 1。进一步刻画了k为奇数时3≤r≤k+12−3和r≥k+1的极值超图，以及k为偶数时3≤r≤k2−2和r≥k+1的极值超图。对于其他几种情况，我们建立了exr（n,Berge-Ln,k）的上界。我们的结果扩展了最近发表的关于Turán berge匹配和线性森林数量的结果。

引用次数: 0

Uniformly sampling random directed hypergraphs with fixed degrees 固定度随机有向超图的均匀抽样

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-05 DOI: 10.1016/j.disc.2025.114961

Yanna J. Kraakman, Clara Stegehuis

Many complex systems show non-pairwise interactions, which can be captured by hypergraphs. In this work, we propose an edge-swapping method to sample random directed hypergraphs with fixed vertex and hyperarc degrees, which can be applied to different classes of directed hypergraphs (containing self-loops, degenerate hyperarcs and/or multi-hyperarcs). We prove that this method indeed samples uniformly from the classes with self-loops and multi-hyperarcs, and that the method may not sample uniformly from classes without self-loops, or with self-loops and degenerate hyperarcs but without multi-hyperarcs. We present a partial result on the class with self-loops, but without degenerate hyperarcs or multi-hyperarcs.

许多复杂系统表现出非成对相互作用，这可以通过超图来捕获。在这项工作中，我们提出了一种边交换方法来采样具有固定顶点和超弧度的随机有向超图，该方法可以应用于不同类别的有向超图（包含自环，退化超弧和/或多超弧）。我们证明了该方法确实能从具有自环和多超弧的类中均匀采样，而该方法不能从没有自环的类中均匀采样，也不能从具有自环和退化超弧但没有多超弧的类中均匀采样。我们给出了有自循环但没有退化超弧或多超弧的类的部分结果。

引用次数: 0

Kempe equivalence of 4-colorings of graphs on non-orientable surfaces 非定向曲面上图的四色的Kempe等价

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-16 DOI: 10.1016/j.disc.2026.115007

Naoki Matsumoto

Two vertex colorings of a graph are Kempe equivalent if they can be transformed into each other by a sequence of Kempe changes which interchange the colors used on a component of the subgraph induced by two color classes. Fisk showed that every two vertex 4-colorings of a 3-colorable triangulation on the sphere are Kempe equivalent, and then Mohar extended this result to any 3-colorable planar graph. Fisk also verified that there are 4-chromatic triangulations on the sphere and 3-colorable triangulations on the torus such that some two 4-colorings of them are not Kempe equivalent. In this paper, we show that every two vertex 4-colorings of a 3-colorable triangulation on the projective plane or the Klein bottle are Kempe equivalent. Our result is best possible in terms of all conditions; 3-colorability, the genus of a non-orientable surface, a triangulation (i.e., it cannot be replaced with a graph).

如果一个图的两个顶点着色可以通过Kempe变换序列相互转换，则它们是Kempe等价的，Kempe变换序列交换了由两个颜色类引起的子图组件上使用的颜色。Fisk证明了球面上3色三角剖分的每两个顶点4色都是Kempe等价的，然后Mohar将这一结果推广到任意3色平面图。Fisk还验证了球面上存在4色三角剖分，环面上存在3色三角剖分，使得其中的一些4色剖分不是Kempe等价的。本文证明了在投影平面或克莱因瓶上的可三色三角剖分的每两个顶点四色都是Kempe等价的。我们的结果在所有条件下都是最好的；3色性，不可定向曲面的属，三角剖分（即不能用图代替）。

引用次数: 0

Antipodality of spherical designs with odd harmonic indices 奇调和指数球形设计的反对性

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-14 DOI: 10.1016/j.disc.2026.114978

Ryutaro Misawa , Akihiro Munemasa , Masanori Sawa

We determine the smallest size of a non-antipodal spherical design with harmonic indices

{1, 3, \dots, 2 m - 1}

to be

2 m + 1

, where m is a positive integer. This is achieved by proving an analogous result for interval designs.

我们确定具有谐波指数{1,3，…，2m−1}的非对对球设计的最小尺寸为2m+1，其中m为正整数。这是通过证明区间设计的类似结果来实现的。

引用次数: 0

s-almost cross-t-intersecting families for finite sets 有限集的s-几乎交叉相交族

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-14 DOI: 10.1016/j.disc.2026.114982

Dehai Liu , Kaishun Wang , Tian Yao

Two families

F

and

G

of k-subsets of an n-set are called s-almost cross-t-intersecting if each member in

F

(resp.

G

) is t-disjoint with at most s members in

G

(resp.

F

). In this paper, we characterize the s-almost cross-t-intersecting families with the maximum product of their sizes. Furthermore, we provide a corresponding stability result after studying the s-almost cross-t-intersecting families which are not cross-t-intersecting.

两个族F和G （n集合的k个子集）被称为s-几乎交叉相交，如果F中的每个元素都对应。G)是t不相交的，在G (p)中最多有s个元素。F)。在本文中，我们用其大小的最大积刻画了s-几乎交叉相交族。此外，我们在研究了不相交的s-几乎相交族后，给出了相应的稳定性结果。

引用次数: 0

Some frustrating questions on dimensions of products of posets 关于偏序集积维数的一些令人沮丧的问题

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-21 DOI: 10.1016/j.disc.2026.115002

George M. Bergman

The definition of the dimension of a poset is recalled. For a subposet P of a direct product of

d > 0

chains, and an integer

n > 0

, a condition is developed which implies that for any family of n chains

{(T_{j})}_{j \in n}

, one has

\dim (P \times \prod_{j \in n} T_{j}) \leq d

. Applications are noted.

Open questions, old and new, on dimensions of product posets are stated, and some other numerical invariants of posets that seem useful for studying these questions are developed. Some variants of the concept of the dimension of a poset from the literature are also recalled.

In a final section, independent of the other results, it is noted that by the compactness theorem of first-order logic, an infinite poset P has finite dimension d if and only if d is the supremum of the dimensions of its finite subposets.

回顾偏序集维数的定义。对于d>；0链的直积P与整数n>；0，给出了一个条件，该条件表明对于任意n个链（Tj）j∈n的族，有dim (px∏j∈nTj)≤d。应用程序被记录。提出了关于积序集维数的新老开放问题，并给出了对研究这些问题有用的其它一些积序集的数值不变量。还回顾了文献中偏序集维数概念的一些变体。在最后一节中，独立于其他结果，我们注意到，通过一阶逻辑的紧性定理，当且仅当d是其有限集合维数的上限时，无限偏集P具有有限维d。

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引用次数: 0

Revisiting d-distance (independent) domination in trees and in bipartite graphs 回顾树和二部图中的d距离（独立）支配

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-05 DOI: 10.1016/j.disc.2025.114972

Csilla Bujtás , Vesna Iršič Chenoweth , Sandi Klavžar , Gang Zhang

The d-distance p-packing domination number

γ_{d}^{p} (G)

of G is the minimum size of a set of vertices of G which is both a d-distance dominating set and a p-packing. In 1994, Beineke and Henning conjectured that if

d \geq 1

and T is a tree of order

n \geq d + 1

, then

γ_{d}^{1} (T) \leq \frac{n}{d + 1}

. They supported the conjecture by proving it for

d \in {1, 2, 3}

. In this paper, it is proved that

γ_{d}^{1} (G) \leq \frac{n}{d + 1}

holds for any bipartite graph G of order

n \geq d + 1

, and any

d \geq 1

. Trees T for which

γ_{d}^{1} (T) = \frac{n}{d + 1}

holds are characterized. It is also proved that if T has ℓ leaves, then

γ_{d}^{1} (T) \leq \frac{n - ℓ}{d}

(provided that

n - ℓ \geq d

), and

γ_{d}^{1} (T) \leq \frac{n + ℓ}{d + 2}

(provided that

n \geq d

). The latter result extends Favaron's theorem from 1992 asserting that

γ_{1}^{1} (T) \leq \frac{n + ℓ}{3}

. In both cases, trees that attain the equality are characterized and relevant conclusions for the d-distance domination number of trees derived.

G的d距离p-填料支配数γdp(G)是G的一个顶点集的最小大小，该顶点集既是d距离支配集又是p-填料。1994年，Beineke和Henning推测，如果d≥1且T是n阶≥d+1的树，则γd1(T)≤nd+1。他们通过证明d∈{1,2,3}来支持这个猜想。本文证明了γd1(G)≤nd+1对任意阶n≥d+1的二部图G和任意d≥1成立。对γd1(T)=nd+1成立的树T进行表征。还证明了如果T有r个叶，则γd1(T)≤n−r d（假设n−r≥d）， γd1(T)≤n+ r d+2（假设n≥d）。后一个结果推广了1992年的Favaron定理，断言γ11(T)≤n+ l3。在这两种情况下，都对达到相等的树进行了表征，并得出了树的d距离支配数的相关结论。

引用次数: 0

Determinantally equivalent nonzero functions 行列式等价的非零函数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-29 DOI: 10.1016/j.disc.2026.115021

Harry Sapranidis Mantelos

<div><div>We study the problem raised in Marco Stevens (2021) [20] concerning the extension of its main result to the more general (potentially non-symmetric) setting. We construct a counterexample disproving the conjecture proposed in the paper, and subsequently solve it under some additional minor assumptions that preclude such counterexamples.</div><div>The problem is plainly stated as follows: Let Λ be a set and <math><mi>F</mi></math> a field, and suppose that <math><mi>K</mi><mo>,</mo><mi>Q</mi><mo>:</mo><msup><mrow><mi>Λ</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>→</mo><mi>F</mi></math> are two functions such that for any <math><mi>n</mi><mo>∈</mo><mi>N</mi></math> and <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><mi>Λ</mi></math>, the determinants of matrices <math><msub><mrow><mo>(</mo><mi>K</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math> and <math><msub><mrow><mo>(</mo><mi>Q</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>)</mo><mo>)</mo></mrow><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></msub></math> agree. What are all the possible transformations that transform Q into K? In Marco Stevens (2021) [20] the following two were conjectured: <math><mo>(</mo><mi>T</mi><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>f</mi><mo>(</mo><mi>y</mi><mo>,</mo><mi>x</mi><mo>)</mo></math>; and <math><mo>(</mo><mi>T</mi><mi>f</mi><mo>)</mo><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mi>g</mi><mo>(</mo><mi>x</mi><mo>)</mo><mi>g</mi><msup><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup><mi>f</mi><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math> for some nowhere-zero function g. In the same paper, this conjectured classification is verified in the case of symmetric functions K and Q. By extending the graph-theoretic techniques of the paper, we show that under some surprisingly simple and natural conditions the conjecture remains valid even with the symmetry constraints relaxed.</div><div>By taking Λ finite, the above problem, furthermore, reduces to that between two square matrices investigated in [Raphael Loewy, Principal minors and diagonal similarity of matrices, Linear Algebra and its Applications 78 (1986), 23–64]. Hence, o

我们研究了Marco Stevens(2021)[20]中提出的问题，将其主要结果扩展到更一般（潜在非对称）的设置。我们构造了一个反例来反驳本文提出的猜想，并随后在一些排除此类反例的附加小假设下解决了它。设Λ是一个集合，F是一个域，设K，Q：Λ2→F是两个函数，使得对于任意n∈n和x1，x2，…，xn∈Λ，矩阵（K(xi,xj)）的行列式（K(xi,xj)）1≤i，j≤n和（Q(xi,xj)）1≤i，j≤n一致。把Q变成K的所有可能的变换是什么？在Marco Stevens(2021)[20]中推测了以下两点：（Tf）(x,y)=f(y,x)；和（Tf）(x,y)=g(x)g(y) - 1f（x,y）对于某个无零函数g。在同一篇文章中，在对称函数K和q的情况下验证了这个猜想分类。通过扩展本文的图论技术，我们证明了在一些非常简单和自然的条件下，即使对称约束放松，猜想仍然有效。通过取Λ有限，进一步将上述问题简化为[Raphael Loewy，矩阵的主次和对角相似性，线性代数及其应用78(1986)，23-64]中研究的两个方阵之间的问题。因此，本文给出了一个简单的非线性代数证明，它只使用了一些初等组合和三个简单的3圈和4圈代数恒等式。

引用次数: 0

Around the number of trees in distance-hereditary graphs 关于距离遗传图中树的数目

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-20 DOI: 10.1016/j.disc.2026.115010

Danila Cherkashin , Pavel Prozorov

Counting the number of spanning trees in specific classes of graphs has attracted growing attention in recent years. In this note, we present unified proofs and generalizations of several results obtained during the 2020s. Our main approach is to study the behavior of the vertex (degree) enumerator polynomial of a distance-hereditary graph under certain graph-theoretical operations. The first result provides a factorization formula applicable to graphs admitting a cut whose edges form a complete bipartite subgraph.

One of the central open problems in this area is Ehrenborg's conjecture, which asserts that a Ferrers–Young graph maximizes the number of spanning trees among all bipartite graphs with the same degree sequence. The second main result of this paper shows the equivalence between Ehrenborg's conjecture and its polynomial version.

近年来，计算特定图类中生成树的数量引起了越来越多的关注。在本文中，我们对本世纪20年代获得的几个结果进行了统一的证明和推广。我们的主要方法是研究距离遗传图的顶点（度）枚举数多项式在某些图论操作下的行为。第一个结果提供了一个适用于有切点的图的因式分解公式，这些切点的边构成了完全二部子图。该领域的一个中心开放问题是Ehrenborg猜想，它断言ferers - young图在所有具有相同次序列的二部图中生成树的数量最大。本文的第二个主要结果证明了Ehrenborg猜想与其多项式版本之间的等价性。

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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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