Discrete Applied Mathematics最新文献

英文中文

Planar graphs without cycles of length 4 or 5 are (7:2)-colorable 没有周期长度为4或5的平面图是（7:2）可着色的

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-19 DOI: 10.1016/j.dam.2025.12.033

Yingqian Wang

Given positive integers

a

and

b

with

a \geq b

, a graph

G

is said to be

(a : b)

-colorable, if we can assign

b

colors from color set

{1, 2, \dots, a}

to each vertex of

G

so that adjacent vertices receive no common colors. Let

S

denote the family of planar graphs without cycles of length 4 or 5. It is known that there are graphs in

S

which are not

(3 : 1)

-colorable (Cohen-Addad et al., 2017), and every graph in

S

is list

(11 : 3)

-colorable (Dvořák and Hu, 2019). This paper shows that every graph in

S

(7 : 2)

-colorable. This gives a new relaxation for refuted Steinberg’s conjecture in the setting of

(a : b)

-coloring.

给定a≥b的正整数a和b，图G是（a:b）可着色的，如果我们可以从颜色集{1,2，…，a}中为G的每个顶点分配b种颜色，使得相邻的顶点没有共同的颜色。设S表示无环长为4或5的平面图族。已知S中存在非（3:1）可着色的图（Cohen-Addad et al., 2017）， S中的每个图都是list（11:3）可着色的（Dvořák and Hu, 2019）。本文证明了S中的每一个图都是（7:2）可着色的。这给在(a:b)-着色集合中被驳倒的Steinberg猜想提供了一个新的松弛。

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引用次数: 0

Graph problems and monotone classes 图问题与单调类

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-19 DOI: 10.1016/j.dam.2025.12.029

Vadim Lozin

We study properties of graph classes that are closed under taking subclasses, such as boundedness of graph parameters or polynomial-time solvability of algorithmic problems. In the universe of minor-closed classes of graphs, any such property can be described by a set of minimal classes that do not possess the property, because the minor relation is a well-quasi-order. This, however, is not the case for the subgraph relation, implying that in the universe of monotone classes, which extends the family of minor-closed classes, the existence of minimal classes is not guaranteed. To overcome this difficulty, we employ the notion of boundary classes. Together with minimal classes they play a critical role for classes defined by finitely many forbidden subgraphs. In the present paper, we identify several levels in the hierarchy of monotone classes and describe respective critical classes. In particular, we show that a finitely-defined monotone class

X

has bounded chromatic number, degeneracy, functionality and admits an implicit representation if and only if

X

excludes a forest. We also show that

X

has bounded tree-, clique- and twin-width and admits polynomial-time solutions for a variety of algorithmic problems if and only if

X

excludes a tripod, i.e. a subcubic forest every connected component of which has at most one cubic vertex. The last result, however, does not apply to the Hamiltonian cycle problem. Towards identifying critical classes for this problem we determine complexity of the Hamiltonian cycle problem in some monotone classes.

我们研究了图类在取子类时闭合的性质，如图参数的有界性或算法问题的多项式时间可解性。在图的小闭类的宇宙中，任何这样的性质都可以用一组不具有该性质的最小类来描述，因为小关系是一个良拟序。然而，对于子图关系却不是这样，这意味着在单调类的宇宙中，它扩展了小闭类族，最小类的存在性是不能保证的。为了克服这个困难，我们采用了边界类的概念。它们与最小类一起，对于由有限多个禁止子图定义的类起着关键作用。在本文中，我们在单调类的层次中识别了几个层次，并描述了各自的临界类。特别地，我们证明了有限定义单调类X具有有界色数、简并性、泛函性，并且当且仅当X排除森林时允许隐式表示。我们还证明了X具有有界的树宽度、团宽度和双宽度，并且当且仅当X不包含三脚架，即每个连通成分最多有一个三次顶点的次三次森林时，各种算法问题都有多项式时间解。然而，最后的结果并不适用于哈密顿循环问题。为了确定这个问题的临界类，我们确定了一些单调类中哈密顿循环问题的复杂度。

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引用次数: 0

Semi-total domination in unit disk graphs and general graphs 单位盘图和一般图的半全支配

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.dam.2025.12.016

Sasmita Rout , Gautam Kumar Das

Let

G = (V, E)

be a simple undirected graph with no isolated vertex. A set

D \subseteq V

is a dominating set if each vertex

u \in V

is either in

D

or is adjacent to a vertex

v \in D

. A set

D_{t 2} \subseteq V

is called a semi-total dominating set if

(i)

D_{t 2}

is a dominating set, and

(i i)

for every vertex

u \in D_{t 2}

, there exists another vertex

v \in D_{t 2}

such that the distance between

u

and

v

G

is at most 2. Given a graph

G

, the semi-total domination problem finds a semi-total dominating set of minimum size. This problem is known to be NP-complete for general graphs and remains NP-complete for some special graph classes, such as planar, split, and chordal bipartite graphs. In this paper, we demonstrate that the problem is also NP-complete for unit disk graphs and propose a

6 -

factor approximation algorithm. The algorithm’s running time is

O (n log n)

, where

n

is the number of vertices in the given unit disk graph. In addition, we show that the minimum semi-total domination problem in a graph with maximum degree

Δ

admits a

2 + ln (Δ + 1) -

factor approximation algorithm, which is an improvement over the best-known result

2 + 3 ln (Δ + 1)

设G=（V，E）是一个没有孤立顶点的简单无向图。如果每个顶点u∈V在D中或与顶点V∈D相邻，则集合D是支配集。若(i) Dt2是一个控制集，且（ii）对于每一个顶点u∈Dt2，存在另一个顶点V∈Dt2，且在G中u与V的距离不大于2，则称集合Dt2为半全控制集。给定一个图G，半全支配问题求一个最小大小的半全支配集。对于一般图，这个问题是np完全的，对于一些特殊的图类，如平面图、分割图和弦二部图，这个问题仍然是np完全的。在本文中，我们证明了这个问题对于单位磁盘图也是np完全的，并提出了一个六因子逼近算法。该算法的运行时间为O(nlogn)，其中n为给定单位磁盘图中的顶点数。此外，我们还证明了最大度为Δ的图的最小半全控制问题允许使用2+ln（Δ+1）因子逼近算法，这是对最著名的结果2+3ln（Δ+1）的改进。

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引用次数: 0

An upper bound on path cover number of regular graphs and its application to Hamiltonian cycle in tough graphs 正则图的路径覆盖数上界及其在难图哈密顿循环中的应用

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.dam.2025.12.036

Xiaodan Chen, Xiaoning Yang

The path cover number of a graph

G

is the minimum integer

β

such that

G

contains

β

vertex-disjoint paths that cover all of its vertices. In this paper, we first establish an upper bound on the path cover number for regular graphs. Then we apply this bound to help to derive sufficient conditions for a

t

-tough graph to be Hamiltonian with integer

t \geq 1

, in terms of the edge number of the graph, which improve some known results in the literature. Another key tool we used to derive these sufficient conditions is the (complete) toughness closure lemma due to Hoàng and Robin (2024) and Shan and Tanyel (2025).

图G的路径覆盖数是最小整数β，使得G包含覆盖其所有顶点的不相交路径β。本文首先建立了正则图的路径覆盖数的上界。然后，我们利用这个界，从图的边数出发，得到了t-tough图是整数t≥1的哈密顿算子的充分条件，改进了文献中一些已知的结果。我们用来推导这些充分条件的另一个关键工具是（完全）韧性闭合引理，这是由Hoàng和Robin（2024）以及Shan和Tanyel（2025）得出的。

引用次数: 0

Proof of a conjecture on graph polytope 图多边形上一个猜想的证明

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.dam.2025.12.034

Feihu Liu

Graph polytopes arising from vertex-weighted graphs were first introduced by Bóna, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is palindromic, using Stanley’s reciprocity theorem. Furthermore, we introduce hypergraph polytopes and establish that every simple, connected, unimodular hypergraph polytope is an integer polytope. Additionally, for simple connected uniform hypergraph polytopes, we demonstrate that the numerator polynomial of their Ehrhart series is palindromic.

由顶点加权图产生的图多边形首先由Bóna、Ju和Yoshida提出。利用Stanley互易定理，证明了对于任意简单连通图，其图多面体的Ehrhart级数的分子多项式是回文的一个猜想。进一步引入超图多面体，并证明了每一个简单、连通、单模的超图多面体都是整数多面体。此外，对于简单连通一致超图多边形，我们证明了它们的Ehrhart级数的分子多项式是回文的。

引用次数: 0

The parameterized complexity of Strong Conflict-Free Vertex-Connection Colorability 强无冲突顶点连接着色性的参数化复杂度

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.dam.2025.12.025

Carl Feghali , Hoang-Oanh Le , Van Bang Le

This paper continues the study of a new variant of graph coloring with a connectivity constraint recently introduced by Hsieh et al. (2024). A path in a vertex-colored graph is called conflict-free if there is a color that appears exactly once on its vertices. A connected graph is said to be strongly conflict-free vertex-connection

k

-colorable if it admits a (proper) vertex

k

-coloring such that any two distinct vertices are connected by a conflict-free shortest path. Among others, we show that deciding, for a given graph

G

and an integer

k

, whether

G

is strongly conflict-free vertex-connection

k

-colorable is fixed-parameter tractable when parameterized by the vertex cover number. But under the standard complexity-theoretic assumption

NP ⁄ \subseteq coNP/poly

, deciding, for a given graph

G

, whether

G

is strongly conflict-free vertex-connection 3-colorable does not admit a polynomial kernel, even for bipartite graphs. This kernel lower bound is in stark contrast to the ordinal

k

-coloring problem which is known to admit a polynomial kernel when parameterized by the vertex cover number.

本文继续研究了最近由Hsieh等人（2024）引入的带有连通性约束的图着色的新变体。如果有一种颜色在顶点上只出现一次，则顶点颜色图中的路径称为无冲突。如果连通图允许（适当的）顶点k着色，使得任意两个不同的顶点通过无冲突最短路径连接，则称连通图是强无冲突顶点连接k可着色的。其中，我们证明，对于给定图G和整数k，当用顶点覆盖数参数化时，确定G是否为强无冲突顶点连接k可着色是固定参数可处理的。但在标准复杂性理论假设NP/ coNP/poly下，对于给定图G，判定G是否为强无冲突顶点连接3色，即使对于二部图也不允许多项式核。这个核下界与序数k-着色问题形成鲜明对比，当用顶点覆盖数参数化时，序数k-着色问题承认一个多项式核。

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引用次数: 0

A note on the generalized Turán number of star forests 关于星林广义Turán数目的说明

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.dam.2025.12.032

Yu-Yue Zhang, Jian-Hua Yin

<div><div>The generalized Turán number <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math> is defined to be the maximum number of copies of a complete graph <math><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub></math> in any <math><mi>H</mi></math>-free graph on <math><mi>n</mi></math> vertices. Let <math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math> denote the star on <math><mrow><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow></math> vertices, and let <math><mrow><mi>k</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></mrow></math> denote the disjoint union of <math><mi>k</mi></math> copies of <math><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math>. Gan et al. and Chase determined <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math> for all integers <math><mrow><mi>s</mi><mo>≥</mo><mn>3</mn></mrow></math>, <math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math>. Recently, Liu and Yin further investigated the problem of determining the generalized Turán number of star forests. They determined <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>2</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>s</mi><mo>≥</mo><mn>4</mn></mrow></math>, <math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>3</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>s</mi><mo>≥</mo><mn>5</mn></mrow></math>, <math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></math>. Moreover, they also determined <math><mrow><mtext>ex</mtext><mrow><mo>(</mo><mi>n</mi><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>,</mo><mn>4</mn><msub><mrow><mi>S</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></mrow></mrow></math> for <math><mrow><mi>s</mi><mo>≥</mo><mn>6</mn></mrow></math>, <math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math> and <math><mrow><mi>n</mi><mo>≥</mo><mn>1<

广义的Turán数ex（n,Ks，H）被定义为在任意n个顶点上的无H图中完全图k的最大拷贝数。设sz表示在1个顶点上的星，kz表示sz的k个拷贝的不相交并。Gan et al.和Chase确定了所有整数S≥3、r≥1和n≥1的ex（n,Ks，S）。最近，Liu和Yin进一步研究了确定广义Turán星林数的问题。当s≥4、r≥1、n≥1时确定ex(n,Ks,2S)，当s≥5、r≥1、n≥1时确定ex（n,Ks,3S）。此外，他们还确定了s≥6，r≥1和n≥1时的ex（n,Ks,4S）。然而，确定2≤k≤4和3≤s≤k+1的ex（n,Ks, Ks）的问题似乎是困难和具有挑战性的。本文研究了上述问题，并确定了当s=3时ex（n,Ks,2S）和当3≤s≤4时ex（n,Ks,3S）。此外，我们还确定了3≤s≤5时ex（n,Ks,4S）。

引用次数: 0

On tree–wheel Ramsey numbers 在拉姆齐数字上

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-18 DOI: 10.1016/j.dam.2025.12.007

Yanbo Zhang , Yaojun Chen , Yunqing Zhang

Given two graphs

G

and

H

, the Ramsey number

R (G, H)

is the smallest positive integer

r

such that every graph on

r

vertices contains

G

as a subgraph or its complement contains

H

as a subgraph. Let

T_{n}

denote a tree on

n

vertices, and let

W_{m}

denote a wheel on

m + 1

vertices. Chen et al. (2004) asked for a characterization of those trees

T_{n}

satisfying

R (T_{n}, W_{2 m}) = 2 n - 1

for

m \geq 2

and

n \geq 2 m - 1

. Subsequently, Hafidh and Baskoro (2021) conjectured that if

T_{n}

is a tree with maximum degree at most

n - 2 m + 2

, then

R (T_{n}, W_{2 m}) = 2 n - 1

for

m \geq 2

and

n \geq 2 m + 1

. More recently, Britz et al. (2025) verified this conjecture partially for

m = 4

and large

n

In this note, we resolve the conjecture of Hafidh and Baskoro for all large

n

给定两个图G和H，拉姆齐数R（G，H）是最小的正整数R，使得在R个顶点上的每个图都包含G作为子图或其补包含H作为子图。Tn表示有n个顶点的树，Wm表示有m+1个顶点的轮。Chen等人（2004）要求对m≥2和n≥2m - 1时满足R(Tn,W2m)=2n - 1的树Tn进行表征。随后，Hafidh和Baskoro（2021）推测，如果Tn是最大度不超过n - 2m+2的树，则当m≥2和n≥2m+1时，R(Tn,W2m)=2n - 1。最近，Britz等人（2025）对m=4和大n部分验证了这一猜想。在本文中，我们解决了Hafidh和Baskoro对所有大n的猜想。

引用次数: 0

Equitable greedy scalar equilibria for coalitional semi-cooperative games 联合半合作对策的公平贪婪标量均衡

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-17 DOI: 10.1016/j.dam.2025.11.036

Davoud Foroutannia , Davoud Mahmoudinia

In this paper, we examine coalitional semi-cooperative games in strategic form, where players within each coalition adhere to the equitable preference relation. Equitable preference is a refinement of the Pareto preference relation. It adds to the reflexivity, strict monotonicity, and transitivity of the Pareto preference order the requirements of impartiality and the Pigou–Dalton principle of transfers. We introduce the concept of

C

-equitable, in which

C

is a complete set of coalitions. By generating a finite subset of Pareto optimal solutions, this concept can help decision-makers select the most suitable solution, especially when there is a large number of Pareto equilibria in some real situations.

A scalar equilibrium achieved by maximizing an appropriate utility function over the acceptable joint actions may not be

C

-equitable in many instances. Another goal of this paper is to generate the

C

-equitable equilibria using utility functions. To achieve this, we require the aggregate utility functions to be Schur-convex, in addition to assuming strict increases, to ensure that each scalar equilibrium is a

C

-equitable equilibrium.

The greedy scalar equilibrium is one of the well-known scalar equilibria that assigns actions to players that yield the largest individual payoffs jointly possible. Our final concept is

C

-equitable greedy scalar equilibrium, which incorporates the notions of equitability and greed to select the optimal set of coalitions for a semi-cooperative game.

在本文中，我们研究了联盟半合作博弈的策略形式，其中每个联盟中的参与者都遵守公平偏好关系。公平偏好是对帕累托偏好关系的改进。它增加了帕累托偏好顺序的反思性、严格单调性和传递性、公正性要求和庇古-道尔顿转移原则。我们引入了C-公平的概念，其中C是一个完整的联盟集合。通过生成Pareto最优解的有限子集，该概念可以帮助决策者选择最合适的解，特别是在某些实际情况下存在大量Pareto均衡时。在许多情况下，通过在可接受的联合行动上最大化适当的效用函数来实现的标量平衡可能不是c -公平的。本文的另一个目标是利用效用函数生成c -公平均衡。为了实现这一点，我们要求聚合效用函数是schur -凸的，除了假设严格的增加，以确保每个标量均衡是c -公平均衡。贪心标量均衡是一种著名的标量均衡，它将行动分配给玩家，使其产生最大的个人收益。我们的最后一个概念是c公平贪婪标量均衡，它结合了公平和贪婪的概念，为半合作博弈选择最优的联盟集。

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引用次数: 0

List 3-dynamic colorings of planar graphs 列出3种平面图的动态着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-17 DOI: 10.1016/j.dam.2025.12.020

Wenjuan Wei, Min Chen

r

-dynamic coloring of a graph

G

is a proper vertex coloring such that for every vertex

v \in V (G)

, at least

min {r, d_{G} (v)}

distinct colors appear in its neighborhood

N_{G} (v)

. Given a list assignment

L = {L (v) ∣ v \in V (G)}

, if there exists an

r

-dynamic coloring

π

G

such that

π (v) \in L (v)

for every vertex

v \in V (G)

, then we say that

G

r

-dynamically

L

-colorable. If

G

r

-dynamically

L

-colorable for every list assignment

L

with

| L (v) | \geq k

for every vertex

v \in V (G)

, then

G

is called to be

r

-dynamically

k

-choosable. It is known that every planar graph is 3-dynamically 10-choosable and it has been conjectured that they are 3-dynamically 8-choosable. In this paper, we sure prove that every planar graph is 3-dynamically 9-choosable.

图G的r-动态着色是一个适当的顶点着色，使得对于每个顶点v∈v (G)，在它的邻域NG(v)中至少有min{r，dG(v)}种不同的颜色出现。给定一个列表赋值L={L(v)∣v∈v (G)}，如果G存在一个r-动态着色π，使得π(v)∈L(v)对于每个顶点v∈v (G)，则我们说G是r-动态L-可着色的。如果对于每个顶点v∈v (G)，对于每一个|L(v)|≥k的列表赋值L， G是r-动态L可色的，则称G是r-动态k可选的。已知每个平面图都是3-动态可选的，并推测它们是3-动态可选的。在本文中，我们肯定地证明了每一个平面图都是动态可选的。

引用次数: 0

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