Discrete Applied Mathematics最新文献

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A quotient-lifting approach to the Hamiltonicity of the cylindrical 5-puzzle graph 圆柱五谜图哈密顿性的提商方法

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-06-15 Epub Date: 2026-01-29 DOI: 10.1016/j.dam.2026.01.031

Taizo Sadahiro

In this short note, we construct an explicit Hamiltonian cycle in the state graph of the 5-puzzle on a cylindrical 2 × 3 grid, a graph with 720 vertices. The cycle is described by a short symbolic sequence of 48 moves over the alphabet

{L, R, V}

, repeated 15 times, which can be verified directly. We also find a shorter 24-move sequence whose repetition yields a 2-cycle cover, which can be spliced into a Hamiltonian path. These constructions arise naturally from a general method: lifting Hamiltonian cycles from a quotient graph under the action of the puzzle’s symmetry group. The method produces compact, human-readable cycle encoding.

在这篇简短的笔记中，我们在一个有720个顶点的圆柱形2 × 3网格上的5谜题的状态图中构造了一个显式的哈密顿循环。这个循环是用一个简短的符号序列来描述的，在字母表{L，R，V}上移动48次，重复15次，这可以直接验证。我们还发现了一个更短的24步序列，它的重复产生一个2周期的覆盖，它可以拼接成一个哈密顿路径。这些构造是由一种一般的方法自然产生的：在谜题对称群的作用下，从商图中提出哈密顿环。该方法生成紧凑的、人类可读的循环编码。

引用次数: 0

Edge general position in graphs: Graph products, integer linear programming and some applications 图中的边一般位置：图积、整数线性规划及一些应用

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-06-15 Epub Date: 2026-01-27 DOI: 10.1016/j.dam.2026.01.008

Zahra Hamed-Labbafian , Michael A. Henning , Mostafa Tavakoli

An edge general position set in a graph

G

is a set of edges

X \subseteq E (G)

where no three distinct edges in

X

lie on a common shortest path. The edge general position number, denoted gp_e is the maximum cardinality of an edge general position set in

G

. In this study, we explore the edge general position number for various graph products, including the hierarchical product, corona product, and edge corona product. Additionally, we propose an integer linear programming model to address the edge general position problem.

图G中的边一般位置集是X⊥E(G)的边集，其中X中没有三条不同的边位于公共最短路径上。边一般位置数，记为gpe，是g中边一般位置集的最大基数。在本研究中，我们探讨了各种图积的边一般位置数，包括层次积、电晕积和边电晕积。此外，我们提出了一个整数线性规划模型来解决边缘一般位置问题。

引用次数: 0

Cops and robbers on multi-layer graphs 多层图上的警察和劫匪

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-05-31 Epub Date: 2026-01-28 DOI: 10.1016/j.dam.2026.01.017

Jessica Enright , Kitty Meeks , William Pettersson , John Sylvester

We generalise the popular cops and robbers game to multi-layer graphs, where each cop and the robber are restricted to a single layer (or set of edges). We demonstrate that initial intuition about the best way to allocate cops to layers is not always correct, and prove that the multi-layer cop number is neither bounded from above nor below by any increasing function of the cop numbers of the individual layers. We show that it is

NP

-hard to decide if

k

cops are sufficient to catch the robber, even if every cop layer is a tree and a set of isolated vertices. However, we give a polynomial time algorithm to determine if

k

cops can win when the robber layer is a tree. Additionally, we investigate a question of worst-case divisions of a simple graph into layers: given a simple graph

G

, what is the maximum number of cops required to catch a robber over all multi-layer graphs where each edge of

G

is in at least one layer and all layers are connected? For cliques, suitably dense random graphs, and graphs of bounded treewidth, we determine this parameter up to multiplicative constants. Lastly we consider a multi-layer variant of Meyniel’s conjecture, and show the existence of an infinite family of graphs whose multi-layer cop number is bounded from below by a constant times

n / log n

, where

n

is the number of vertices in the graph.

我们将流行的警察和强盗游戏推广到多层图，其中每个警察和强盗都被限制在一个单层（或一组边）。我们证明了关于将警察分配到各层的最佳方法的初始直觉并不总是正确的，并证明了多层警察数量既不受各个层警察数量的任何增加函数的上下限制。我们证明，即使每个警察层都是一棵树和一组孤立的顶点，决定k个警察是否足以抓住抢劫犯也是np困难的。然而，我们给出了一个多项式时间算法来确定当强盗层是树时k个警察是否可以获胜。此外，我们还研究了一个简单图分层的最坏情况划分问题：给定一个简单图G，在所有多层图中（其中G的每条边至少在一层中，并且所有层都是连通的），抓捕抢劫犯所需的警察的最大数量是多少？对于团、适当密集的随机图和有界树宽的图，我们确定这个参数直至乘法常数。最后，我们考虑了Meyniel猜想的一个多层变体，并证明了一个无限族图的存在性，其多层cop数由一个常数乘以n/logn，其中n为图中的顶点数。

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引用次数: 0

Extremal oriented graphs avoiding 1-subdivision of an in-star 极值定向图避免了内星的1-细分

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-05-31 Epub Date: 2026-01-27 DOI: 10.1016/j.dam.2026.01.011

Zejun Huang, Chenxi Yang

An oriented graph is a digraph obtained from an undirected graph by choosing an orientation for each edge. Given a positive integer

n

and an oriented graph

F

, the oriented Turán number

e x_{o r i} (n, F)

is the maximum number of arcs in an

F

-free oriented graph of order

n

. In this paper, we investigate the oriented Turán number

e x_{o r i} (n, \vec{S_{k, 1}})

, where

\vec{S_{k, 1}}

is the 1-subdivision of the in-star of order

k + 1

. We determine

e x_{o r i} (n, \vec{S_{k, 1}})

for

k = 2, 3

as well as the extremal oriented graphs. For

k \geq 4

, we establish a lower bound and an upper bound on

e x_{o r i} (n, \vec{S_{k, 1}})

有向图是由无向图通过为每条边选择一个方向而得到的有向图。给定一个正整数n和一个有向图F，有向Turán数exori（n，F）是一个n阶无F有向图中弧的最大数目。本文研究了有向Turán数exori（n,Sk，1分），其中Sk，1分是k+1阶in-star的1细分。我们确定了k=2,3以及极值取向图的exori（n,Sk,1∈）。对于k≥4，我们建立了exori（n,Sk,1∈）上的下界和上界。

引用次数: 0

S-packing chromatic critical graphs s填充色临界图

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-05-31 Epub Date: 2026-01-22 DOI: 10.1016/j.dam.2026.01.024

Gülnaz Boruzanlı Ekinci , Csilla Bujtás , Didem Gözüpek , Sandi Klavžar

For a non-decreasing sequence of positive integers

S = (s_{1}, s_{2}, \dots)

, the

S

-packing chromatic number of a graph

G

is denoted by

χ_{S} (G)

. In this paper,

χ_{S}

-critical graphs are introduced as the graphs

G

such that

χ_{S} (H) < χ_{S} (G)

for each proper subgraph

H

G

. Several families of

χ_{S}

-critical graphs are constructed, and 2- and 3-colorable

χ_{S}

-critical graphs are presented for all packing sequences

S

, while 4-colorable

χ_{S}

-critical graphs are found for most of

S

. Cycles which are

χ_{S}

-critical are characterized under different conditions. It is proved that for any graph

G

and any edge

e \in E (G)

, the inequality

χ_{S} (G - e) \geq χ_{S} (G) / 2

holds. Moreover, in several important cases, this bound can be improved to

χ_{S} (G - e) \geq (χ_{S} (G) + 1) / 2

. The sharpness of the bounds is also discussed. Along the way an earlier result on

χ_{S}

-vertex-critical graphs is supplemented.

对于非递减的正整数序列S=（s1,s2，…），图G的S填充色数用χS(G)表示。本文将χS临界图作为图G引入，对G的每个固有子图H都构造了χS(H)<χS(G)。构造了几类χS临界图，对所有填充序列S都给出了2色和3色的χS临界图，对大多数S循环都得到了4色的χS临界图，并在不同条件下对其进行了χS临界表征。证明了对于任意图G和任意边e∈e (G)，不等式χS(G−e)≥χS(G)/2成立。此外，在一些重要的情况下，这个界限可以改进为χS(G−e)≥(χS(G)+1)/2。讨论了边界的清晰度。在此过程中，对先前关于χ s -顶点临界图的结果进行了补充。

引用次数: 0

The generalized 3-connectivity of BCCC data center networks BCCC数据中心网络的广义3连通性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-05-31 Epub Date: 2026-02-10 DOI: 10.1016/j.dam.2026.02.007

Baohua Niu , Yan Wang , Hai Liu , Baolei Cheng , Jianxi Fan

As the core infrastructure for technologies such as cloud computing, big data and artificial intelligence, data center networks (DCNs) realize efficient collaborative communication among massive servers, storage devices and terminal nodes through effective interconnection, reliable transmission and intelligent management mechanisms. In typical application scenarios like distributed computing, real-time business response and elastic resource expansion, the demand for multi-node collaborative communication is becoming more and more prominent. The traditional connectivity theory is limited to the analysis of connectivity between two nodes, which is difficult to adapt to the complex multi-node collaboration scenarios in data centers. To break this limitation, the concept of generalized connectivity based on the internally disjoint Steiner trees emerges as an important indicator to measure the connection strength among multiple nodes, which plays a significant role in characterizing the reliability and fault tolerance among multiple nodes. BCube connected crossbars (abbreviated as BCCC), a new server-centric DCN, exhibits superior network capabilities, including great scalability, low communication latency and high robustness to component failures. This work addresses the issue of packing internally disjoint Steiner trees in the logic graph of BCCC. Specifically, we determine the largest number of internally disjoint Steiner trees that connect arbitrary three nodes in the logic graph of BCCC, thereby deriving precise values of generalized 3-connectivity.

数据中心网络作为云计算、大数据、人工智能等技术的核心基础设施，通过有效互联、可靠传输和智能管理机制，实现海量服务器、存储设备和终端节点之间的高效协同通信。在分布式计算、实时业务响应、弹性资源扩展等典型应用场景中，对多节点协同通信的需求越来越突出。传统的连通性理论仅限于分析两个节点之间的连通性，难以适应数据中心复杂的多节点协作场景。为了打破这一局限，基于内部不连通Steiner树的广义连通性概念作为衡量多节点间连接强度的重要指标出现，对表征多节点间的可靠性和容错能力具有重要作用。BCube连接交叉条（简称BCCC）是一种以服务器为中心的新型DCN，具有卓越的网络功能，包括高可扩展性、低通信延迟和对组件故障的高鲁棒性。本文研究了BCCC逻辑图中内部不相交斯坦纳树的填充问题。具体地说，我们确定了连接BCCC逻辑图中任意三个节点的内部不相交Steiner树的最大数，从而得到广义3连通性的精确值。

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引用次数: 0

Disjunctive domination in maximal outerplanar graphs 极大外平面图的析取支配

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-05-31 Epub Date: 2026-01-19 DOI: 10.1016/j.dam.2025.12.056

Michael A. Henning , Paras Vinubhai Maniya , Dinabandhu Pradhan

A disjunctive dominating set of a graph

G

is a set

D \subseteq V (G)

such that every vertex in

V (G) ∖ D

has a neighbor in

D

or has at least two vertices in

D

at distance 2 from it. The disjunctive domination number of

G

, denoted by

γ_{2}^{d} (G)

, is the minimum cardinality among all disjunctive dominating sets of

G

. In this paper, we show that if

G

is a maximal outerplanar graph of order

n \geq 7

with

k

vertices of degree 2, then

γ_{2}^{d} (G) \leq ⌊ \frac{2}{9} (n + k) ⌋

, and this bound is sharp.

图G的一个析取支配集是一个集D⊥V(G)，使得V(G) × D中的每个顶点在D中有一个邻居，或者在距离为2的距离上至少有两个顶点在D中。G的析取支配数，用γ2d(G)表示，是G的所有析取支配集中最小的cardinality。本文证明了如果G是一个n≥7阶的极大外平面图，有k个顶点为2度，则γ2d(G)≤⌊29（n+k）⌋，且该界是尖锐的。

引用次数: 0

Refinements of Combinatorial Nullstellensatz via polynomial supports 基于多项式支持的组合零态的改进

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-05-31 Epub Date: 2026-01-21 DOI: 10.1016/j.dam.2026.01.025

Wei Cao

Nica introduced the notion of nullity for finite sets and utilized it to refine the Combinatorial Nullstellensatz over structured grids. Inspired by Nica’s approach, we replace the concept of nullity with the support of polynomials, thereby obtaining further refinements of the Combinatorial Nullstellensatz.

Nica引入了有限集的零性概念，并利用它来改进结构化网格上的组合零性。受Nica方法的启发，我们用多项式的支持取代了零的概念，从而得到了组合零的进一步改进。

引用次数: 0

Games on deBruijn graphs and cycle means 德布鲁因图和循环方法上的博弈

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-05-31 Epub Date: 2026-01-24 DOI: 10.1016/j.dam.2025.12.060

Nadejda Drenska

deBruijn graphs are widely used in genomics and computer science. In this paper we present a novel approach to finding weights on edges of doubly weighted deBruijn graphs. Given any fixed set of weights on vertices, we use a repeated two-person zero-sum game to find weights on edges so that every cycle on the deBruijn graph has the same average weight, providing explicit formulas. This approach uses minimax optimal strategies of the players. Once the weights on the edges are determined, we observe that they correspond to solving a set of linear equations with as many equations as there are cycles. This is very surprising, because there are many more cycles than unknowns. Moreover we analyze other, related games on graphs.

德布鲁因图广泛应用于基因组学和计算机科学。本文提出了一种求双重加权德布鲁因图边权的新方法。给定任意一组固定的顶点权值，我们使用一个重复的二人零和博弈来找到边上的权值，这样deBruijn图上的每个循环都有相同的平均权值，并提供明确的公式。这种方法使用了参与者的最小最大最优策略。一旦确定了边的权值，我们观察到它们对应于求解一组线性方程，其中有多少个循环就有多少个方程。这是非常令人惊讶的，因为有更多的周期比未知的。此外，我们在图表上分析了其他相关的游戏。

引用次数: 0

Covering a supermodular-like function in a mixed hypergraph 覆盖混合超图中的类超模函数

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2026-05-31 Epub Date: 2026-01-22 DOI: 10.1016/j.dam.2026.01.023

Hui Gao

In this paper, we solve a conjecture by Szigeti in [Matroid-rooted packing of arborescences], which characterizes mixed hypergraphs

F = (V, E \cup A)

for which there exists an orientation

\vec{E}

E

such that

e_{\vec{E} \cup A} (P) \geq \sum_{X \in P} h (X) - b (\cup P)

for every subpartition

P

V

, where

h

is an integer-valued, intersecting supermodular function on

V

and

b

a submodular function on

V

. As a corollary, another conjecture in the same paper is confirmed, which characterizes mixed hypergraphs admitting a packing of mixed hyperarborescences such that their roots form a basis in a given matroid, each vertex

v

belongs to exactly

k

of them and is the root of at least

f (v)

and at most

g (v)

of them.

本文用Szigeti在[树形的矩阵根填充]中求解了一个猜想，该猜想刻画了混合超图F=(V,E∪a)，其中对于V的每一子划分P存在一个方向E∈E，使得E∈∪a (P)≥∑X∈Ph(X)−b(∪P)，其中h是V上的一个整数值相交超模函数，b是V上的一个子模函数，作为一个推论，证实了同一论文中的另一个猜想。这是混合超图的特征，它允许混合超树序列的填充，使得它们的根形成给定矩阵中的一组基，每个顶点v恰好属于其中的k个顶点，并且是至少f(v)最多g(v)的根。

引用次数: 0

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