Discrete Applied Mathematics最新文献

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Even cycle decompositions of the line graphs of cubic graphs 三次图的线形图的连循环分解

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-09 DOI: 10.1016/j.dam.2025.12.052

Chuixiang Zhou, Yuxi Zou

An even cycle decomposition of a graph is defined as a partition of its edges into cycles of even length. Let

G

be a 2-connected cubic graph. Markström conjectured that the line graph

L (G)

G

admits an even cycle decomposition. Suppose that

F = C_{1} \cup C_{2} \cup \dots \cup C_{k}

is a 2-factor of

G

, where each

C_{i}

(

1 \leq i \leq k

) is a cycle. Let

G_{F}

be the graph obtained from

G

by contracting each

C_{i}

(

1 \leq i \leq k

) to a vertex and deleting the resulting loops and parallel edges. In this paper, we prove that Markström’s conjecture is true if

G_{F}

is isomorphic to a tree for some 2-factor

F

G

图的偶环分解被定义为将图的边划分为偶数长度的环。设G是一个二连通三次图。Markström推测G的线形图L(G)允许偶循环分解。设F=C1∪C2∪⋯∪Ck是G的一个2因子，其中每个Ci（1≤i≤k）是一个循环。设GF为由G将每个Ci（1≤i≤k）缩并到一个顶点，并删除由此产生的环路和平行边得到的图。本文证明了如果GF同构于某2因子F (G)的树，则Markström的猜想是成立的。

引用次数: 0

Polarity on H-split graphs h -分裂图的极性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-08 DOI: 10.1016/j.dam.2025.12.065

Fernando Esteban Contreras-Mendoza, César Hernández-Cruz

A partition

(A, B)

of the vertex set of a graph

G

is called a polar partition of

G

when

G [A]

and

\bar{G [B]}

are complete multipartite graphs. If

(A, B)

is a polar partition of

G

in which

G [A]

and

\bar{G [B]}

have at most

s

and

k

parts, respectively, then

(A, B)

is an

(s, k)

-polar partition of

G

, and

G

is said to be an

(s, k)

-polar graph. A graph is said to be unipolar (monopolar) if its vertex set admits a polar partition

(A, B)

such that

A

is a clique (an independent set, respectively). A graph admitting a

(1, 1)

-polar partition is usually called a split graph.

Naturally, most problems related to polar partitions are trivial on split graphs, even when some of them are very hard in general. In this work, we present some results related to polar partitions on two graph classes generalizing split graphs. Our main results include efficient algorithms to decide whether graphs in these classes admit such partitions. We also establish upper bounds on the order of minimal

(s, k)

-polar obstructions for these families, for any

s

and

k

(possibly

s = \infty

k = \infty

当G[A]和G[B]¯是完全多部图时，图G顶点集的划分（A，B）称为G的极划分。如果（A，B）是G的极分割，其中G[A]和G[B]¯分别最多有s和k个部分，则（A，B）是G的（s,k）极分割，G称为（s,k）极图。如果一个图的顶点集允许一个极分割（A，B），使得A是一个团（分别是一个独立的集合），那么这个图就是单极的（单极的）。具有（1,1）极分割的图通常称为分割图。当然，大多数与极坐标划分相关的问题在分割图上都是微不足道的，即使其中一些问题通常非常困难。在本文中，我们给出了关于两个图类的极坐标划分的一些结果。我们的主要成果包括有效的算法来决定这些类中的图是否允许这样的分区。对于任意s和k（可能是s=∞或k=∞），我们也建立了这些族的极小（s,k）极障碍的上界。

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引用次数: 0

On the (total) domination in subdivision graphs 关于细分图中的（总）支配

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-08 DOI: 10.1016/j.dam.2025.12.055

Esteban De Jesús Sánchez , Abel Cabrera-Martínez , Ismael Rios-Villamar , José M. Sigarreta

In this article, we study the total domination number and the

k

-domination number of the subdivision graph operator

S (G)

for simple graphs

G

. First, we obtain a closed formula for the total domination number of this well-known graph operator. Additionally, for

k \geq 2

, we obtain the exact value of the

k

-domination number of

S (G)

. The picture is notably different when considering the case

k = 1

, i.e., the domination number. In this case, we obtain some general bounds on the domination number of

S (G)

, which we further improve when

G

is a tree. Finally, we provide closed formulas for the domination number of the subdivision graph of some well-known composite graphs.

本文研究了简单图G的细分图算子S(G)的总支配数和k-支配数。首先，我们得到了这个著名图算子的总支配数的封闭公式。此外，当k≥2时，我们得到了S(G)的k支配数的确切值。当考虑k=1（即支配数）的情况时，情况明显不同。在这种情况下，我们得到了S(G)的支配数的一般界，当G是树时，我们进一步改进了这一界。最后，我们给出了一些著名的复合图的细分图的支配数的封闭公式。

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引用次数: 0

Vertex evaluation of multiplex graphs using Forman curvature 基于Forman曲率的多路图顶点评估

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-08 DOI: 10.1016/j.dam.2025.12.058

Taiki Yamada

The identification of vertices that play a central role in network analysis is a fundamental challenge. Although traditional centrality measures have been extensively employed for this purpose, the increasing complexity of modern networks necessitates the use of sophisticated metrics. The concept of Forman curvature has recently garnered significant attention as a promising approach. We define the Forman curvature for multiplex graphs, which are a category of complex networks characterized by multiple layers of connections between nodes. We then prove the key properties of the Forman curvature in the context of multiplex graphs and show its usefulness in identifying vertices occupying central positions within these networks. Moreover, through a series of comparative experiments with traditional graph features and graph kernels, we demonstrate that the Forman curvature can function as an effective metric for classifying the overall structure of networks.

识别在网络分析中起核心作用的顶点是一个根本性的挑战。虽然传统的中心性度量已被广泛用于此目的，但现代网络的日益复杂需要使用复杂的度量。福尔曼曲率的概念最近作为一种有前途的方法获得了极大的关注。我们定义了多路图的Forman曲率，多路图是一类以节点之间的多层连接为特征的复杂网络。然后，我们在多重图的背景下证明了Forman曲率的关键性质，并展示了它在识别这些网络中占据中心位置的顶点方面的有用性。此外，通过一系列与传统图特征和图核的对比实验，我们证明了Forman曲率可以作为对网络整体结构进行分类的有效度量。

引用次数: 0

Thick Forests 浓密的森林

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-08 DOI: 10.1016/j.dam.2025.12.042

Martin Dyer , Haiko Müller

We consider classes of graphs, which we call thick graphs, that have the vertices of a corresponding thin graph replaced by cliques and the edges replaced by cobipartite graphs. In particular, we consider the case of thick forests, which we show to be the largest class of perfect thick graphs.

Recognising membership of a class of thick

C

-graphs is NP-complete unless the class

C

is triangle-free, so we focus on this case. Even then membership can be NP-complete. However, we show that the class of thick forests can be recognised in polynomial time.

We consider two well-studied combinatorial problems on thick graphs, independent sets and proper colourings. Since determining the independence or chromatic number of a perfect graph is known to be tractable, we examine the complexity of counting all independent sets and colourings in thick forests.

Finally, we consider two parametric extensions to larger classes of thick graphs: where the parameter is the size of the thin graph, and where the parameter is its treewidth.

我们考虑一类图，我们称之为厚图，它们对应的薄图的顶点被团代替，边缘被二部图代替。特别地，我们考虑了浓密森林的情况，我们证明了它是最大的一类完美厚图。除非类C是无三角形的，否则识别一类厚C图的隶属性是np完全的，因此我们关注这种情况。即使这样，成员资格也可能是np完全的。然而，我们证明了厚森林的类别可以在多项式时间内识别。考虑两个研究得很好的关于厚图、独立集和适当着色的组合问题。由于确定完美图的独立性或色数是容易处理的，我们研究了在茂密的森林中计数所有独立集和着色的复杂性。最后，我们考虑两个参数扩展到更大的粗图类：其中参数是瘦图的大小，其中参数是它的树宽度。

引用次数: 0

A note on the second-largest number of dissociation sets in connected graphs 关于连通图中第二多解离集的注释

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-07 DOI: 10.1016/j.dam.2025.12.045

Pingshan Li, Ke Yang, Wei Jin

A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order

n

and characterized the corresponding extremal graphs. They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order

n

and the corresponding extremal graphs. In this paper, we give a positive answer to this question.

如果一个顶点的子集诱导出一个顶点度最多为1的子图，则称为解离集。最近Yuan等人建立了所有n阶连通图的最大解离集数的上界，并刻画了相应的极值图。他们还提出了一个关于在所有n阶连通图和相应的极值图中解离集数量第二的问题。本文对这一问题给出了肯定的回答。

引用次数: 0

Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues 前两个最大的无符号拉普拉斯特征值和的极值图

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-07 DOI: 10.1016/j.dam.2025.12.046

Zi-Ming Zhou , Zhi-Bin Du , Chang-Xiang He

For a graph

G

, let

S_{2} (G)

be the sum of the first two largest signless Laplacian eigenvalues of

G

, and

f (G) = e (G) + 3 - S_{2} (G)

. Very recently, Zhou et al. (2024) proved that

K_{1, n - 1}^{+}

(the star graph with an additional edge) is the unique graph with minimum value of

f (G)

among the graphs on

n

vertices. In this paper, we prove that the vertex-disjoint union of

K_{1, e (G) - 1}^{+}

and possibly some isolated vertices is the unique graph with minimum value of

f (G)

among the graphs with

e (G)

edges.

对于图G，设S2(G)为G的前两个最大的无符号拉普拉斯特征值的和，f(G)=e(G)+3 - S2(G)。最近，Zhou等人（2024）证明了K1，n−1+（附加一条边的星图）是n个顶点图中f(G)值最小的唯一图。本文证明了K1、e(G)−1+和可能的一些孤立顶点的顶点不相交并是具有e(G)条边的图中f(G)值最小的唯一图。

引用次数: 0

Dimensional edge fault-tolerant Hamiltonicity of (folded) hypercubes （折叠）超立方体的空间边容错哈密顿性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-07 DOI: 10.1016/j.dam.2025.12.051

Junqing Cai , Meirun Chen , Cheng-Kuan Lin

The hypercube

Q_{n}

and folded hypercube

F Q_{n}

serve as fundamental interconnection network topologies in parallel computing, valued for their efficient communication and inherent fault tolerance. This paper investigates their resilience to dimensional-edge faults with respect to three critical Hamiltonian properties: Hamiltonicity, Hamiltonian laceability, and hyper Hamiltonian laceability. We establish precise bounds for fault tolerance in these structures, proving that: (1) For

Q_{n}

, both the dimensional-edge fault-tolerant Hamiltonicity and Hamiltonian laceability equal

2^{n - 1} - n

, while hyper Hamiltonian laceability tolerates up to

2^{n - 1} - 2 n + 2

; (2) For

F Q_{n}

, the dimensional-edge fault-tolerant Hamiltonicity is

2^{n} - n

; (3) For odd-dimensional

F Q_{2 n + 1}

, the dimensional-edge fault-tolerant Hamiltonian laceability and hyper Hamiltonian laceability are

2^{2 n + 1} - 2 n - 1

and

2^{2 n + 1} - 4 n

, respectively. These results significantly advance our understanding of fault tolerance in cube-based network topologies and provide rigorous theoretical guarantees for their reliable operation in practical systems.

超立方体Qn和折叠超立方体FQn作为并行计算中基本的互连网络拓扑，以其高效的通信和固有的容错性而受到重视。本文从哈密顿性、哈密顿可缺性和超哈密顿可缺性这三个关键哈密顿性质出发，研究了它们对维边断裂的弹性。我们建立了这些结构的容错性的精确边界，证明了：(1)对于Qn，维边容错哈密顿性和哈密顿可溶性均等于2n−1−n，而超哈密顿可溶性可容性可达2n−1−2n+2；(2)对于FQn，维边容错哈密度为2n−n；(3)对于奇维FQ2n+1，维边容错哈密顿可缺性为22n+1−2n−1，超哈密顿可缺性为22n+1−4n。这些结果极大地促进了我们对基于立方体的网络拓扑容错的理解，并为其在实际系统中的可靠运行提供了严格的理论保证。

引用次数: 0

On some critical Ramsey numbers involving paths 在一些关键的拉姆齐数上

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-07 DOI: 10.1016/j.dam.2025.12.048

Yahui Zhang, Yan Li, Changxiang He

For graphs

F

G

and

H

, let

F \to (G, H)

signify that any red-blue edge-coloring of

F

contains either a red

G

or a blue

H

. The Ramsey number

R (G, H)

is defined as

m i n {r | K_{r} \to (G, H)}

. In this note, we show that

K_{r} ∖ P_{r} \to (G, P_{n})

and

K_{r} ∖ K_{⌈ \frac{n}{2} ⌉ - 1, ⌈ \frac{n}{2} ⌉ - 1} \to (G, P_{n})

under some bounds on

n

, where

r = R (G, P_{n})

and

G

is a given graph with minimum color class of size 1. Our construction of edge-deleted balanced complete bipartite graphs removes more edges than earlier results concerning the critical Ramsey number of paths.

对于图F， G和H，设F→（G，H）表示F的任何红蓝边着色包含一个红色G或一个蓝色H，拉姆齐数R（G，H）定义为min{R |Kr→（G，H）}。在这篇文章中，我们证明了Kr∈Pr→（G,Pn）和Kr∈K≤≤n2²−1，≤n2²−1→（G,Pn）在n上的某些界下，其中r= r (G,Pn)，并且G是一个最小色类大小为1的给定图。我们构造的边删除平衡完全二部图比先前关于路径临界拉姆齐数的结果删除了更多的边。

引用次数: 0

Pair state transfer in tensor product and double cover 张量积和双盖中的对状态转移

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-01-07 DOI: 10.1016/j.dam.2025.12.053

Ming Jiang , Xiaogang Liu , Jing Wang

Quantum state transfer, first introduced by Bose in 2003, is an important physical phenomenon in quantum networks, which plays a vital role in quantum communication and quantum computing. In 2004, Christandl et al. proposed the concept of perfect state transfer on graphs by modeling the quantum network using graphs, and unveiled the feasibility of applying graph theory to quantum state transfer. In 2018, Chen and Godsil proposed the definition of Laplacian perfect pair state transfer on graphs, which is a brilliant generalization of perfect state transfer. In this paper, we investigate the existence of Laplacian perfect pair state transfer in tensor product and double cover of two regular graphs, respectively, and reveal fundamental connections between perfect state transfer and Laplacian perfect pair state transfer. We give necessary and sufficient conditions for the tensor product of two regular graphs to admit Laplacian perfect pair state transfer, where one of the two regular graphs admits perfect state transfer or Laplacian perfect pair state transfer. Additionally, we characterize the existence of Laplacian perfect pair state transfer in the double cover of two regular graphs. By our results, a variety of families of graphs admitting Laplacian perfect pair state transfer can be constructed.

量子态转移是2003年由Bose首次提出的量子网络中的重要物理现象，在量子通信和量子计算中起着至关重要的作用。2004年，Christandl等人利用图对量子网络进行建模，提出了图上完美状态转移的概念，揭示了将图论应用于量子状态转移的可行性。2018年，Chen和Godsil在图上提出了Laplacian完美对状态转移的定义，这是对完美状态转移的一个精彩推广。本文分别研究了两个正则图的张量积和双覆盖中拉普拉斯完美对状态转移的存在性，揭示了完美状态转移与拉普拉斯完美对状态转移之间的基本联系。给出了两个正则图的张量积允许拉普拉斯完美对状态转移的充分必要条件，其中一个正则图允许完全状态转移或拉普拉斯完美对状态转移。此外，我们还刻画了两个正则图的双覆盖上存在拉普拉斯完美对状态转移。根据我们的研究结果，可以构造各种允许拉普拉斯完美对状态转移的图族。

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