Discrete Applied Mathematics最新文献

英文中文

A random forest process with a variable number of giant components in the threshold window 一个随机森林过程，在阈值窗口中具有可变数量的巨大组件

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-17 DOI: 10.1016/j.dam.2025.12.030

Colin Cooper, Tomasz Radzik

Given a graph

G

, and an ordering

π

of its vertices, a permutation forest

F (G, π)

is a spanning forest of

G

whose components are obtained as follows. For each vertex

v

, connect

v

to its first neighbour

w

G

that appears after

v

in the ordering

π

. If we regard this edge

(v, w)

as directed forward, from

v

w

, then each vertex has at most one forward edge, and the components of

F (G, π)

are arborescences. The roots of the components formed by this process are those vertices of

G

with no forward edge in the ordering

π

This paper shows that the permutation forests of the random graphs

G_{n, p}

have a threshold for the emergence of a linear size component around

p = \sqrt{1 / n}

. In contrast to the w.h.p. emergence of a unique giant in

G_{n, p}

, the permutation forest process has the unusual property that, with positive probability, a number of linear size components occur within the threshold window.

给定图G及其顶点的有序π，则置换森林F（G,π）是G的生成森林，其组成可得如下：对于每一个顶点v，把v和它在G中的第一个邻居w连接起来，这个邻居w在v之后以π的顺序出现。如果我们把这条边（v,w）看作是正向的，从v到w，那么每个顶点最多有一条正向边，F（G,π）的分量是树突。这个过程形成的分量的根是那些在π阶中没有前边的G的顶点。本文证明了随机图Gn，p的排列森林在p=1/n附近有一个出现线性大小分量的阈值。与w.h.p.在Gn，p中出现一个独特的巨人相比，排列森林过程具有不同寻常的特性，即以正概率，在阈值窗口内出现许多线性大小的成分。

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引用次数: 0

(t,r)-Broadcast Domination in graphs （t,r）-图中的广播支配

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-16 DOI: 10.1016/j.dam.2025.12.024

Gennaro Cordasco, Luisa Gargano, Adele A. Rescigno

This paper explores a recently introduced graph problem called (

t, r

)-Broadcast Domination. In this problem, a set of “broadcasting” towers transmit a signal of initial strength

t

. This signal’s strength decreases linearly along the edges. The objective is to find the smallest set of broadcasting towers such that every vertex in the graph receives a cumulative signal strength of at least

r

. The (

t, r

)-Broadcast Domination problem, which generalizes the concept of distance domination, has been recently introduced and studied in a few specific class of graphs, like grids and lattices. However, the (

t, r

)-Broadcast Domination problem has not been studied for general graphs. We present the first study of the complexity of this problem in general graphs, including a general approximation algorithm and optimal polynomial-time algorithms for cographs. We also provide an algorithm parameterized by the neighborhood diversity of the input graph and an algorithm parameterized by modular-width and the solution size.

本文探讨了最近引入的一个称为(t,r)-广播支配的图问题。在这个问题中，一组“广播”塔发射初始强度为t的信号。该信号的强度沿边缘呈线性递减。目标是找到最小的广播塔集合，使得图中的每个顶点接收到的累计信号强度至少为r。(t,r)-广播控制问题，它推广了距离控制的概念，最近在一些特定的图类中被引入和研究，如网格和格子。然而，对于一般图，(t,r)-广播支配问题尚未得到研究。我们首次在一般图中研究了这一问题的复杂性，包括图的一般近似算法和最优多项式时间算法。我们还提供了一种以输入图的邻域多样性为参数的算法和一种以模宽度和解大小为参数的算法。

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引用次数: 0

The structural properties of {(2,3),6}-spheres {(2,3)，6}-球的结构性质

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.dam.2025.12.018

Rui Yang, Chengli Liu

{(2, 3), 6} -

sphere

S

is a

6 -

regular planar graph whose faces are only

2 -

length and

3 -

length. Furthermore, it consists of exactly six

2 -

length faces and even number of

3 -

length faces by Euler’s formula. A graph

G

is called cyclically

k -

edge-connected, if it cannot be separated into at least two components such that each contains cycles by deleting fewer than

k

edges. We denote the maximum value of

k \in N

such that

G

is cyclically

k -

edge-connected by

c λ (G)

, then the positive integer

c λ (G)

is referred to as cyclical edge-connectivity of

G

. In this paper, we prove that the cyclical edge-connectivity of

{(2, 3), 6} -

sphere is 4 or 6 or 8, and we give the structures of

{(2, 3), 6} -

spheres with cyclical edge-connectivities 4 and 6, respectively. A face of

G

is called resonant if its boundary is

M -

alternating cycle, where

M

is a perfect matching of

G

. Furthermore, in this paper, we show that every

2 -

length face of a

{(2, 3), 6} -

sphere is resonant.

A{(2,3)，6}-球面S是一个面只有2长和3长的6正则平面图。根据欧拉公式，它由6个2长面和偶数个3长面组成。如果图G不能被分成至少两个分量，使得每个分量通过删除少于k条边来包含循环，则图G被称为循环k边连通。我们指出k∈N的最大值使得G被cλ(G)循环k边连通，则正整数cλ(G)称为G的循环边连通。本文证明了{(2,3)，6}-球的循环边连通性为4或6或8，并给出了{(2,3)，6}-球的循环边连通性分别为4和6的结构。如果边界是M交替循环，则G的面称为共振面，其中M是G的完美匹配，进一步证明了{(2,3)，6}球面的每一个2长面都是共振面的。

引用次数: 0

Spectral extrema of graphs: Forbidden linear forests and non-bipartite graphs 图的谱极值：禁止线性森林与非二部图

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.dam.2025.12.017

Xuelian Mao, Zhenyu Ni, Ming-Zhu Chen

For a set of graphs

F

, a graph

G

is called

F

-free if it does not contain any graph in

F

as a subgraph. Let SPEX

(n, F)

denote the graphs with the maximum spectral radius among all

F

-free graphs of order

n

. In this paper, for any non-bipartite graph

H

, we give some characterizations for the graphs in SPEX

(n, {⋃_{i = 1}^{ℓ} P_{k_{i}}, H})

for sufficiently large

n

, where

ℓ \geq 1

k_{1} \geq \dots \geq k_{ℓ} \geq 2

, and there exists at least one

k_{i}

not equal to 3. As an application, we completely characterize the graphs in SPEX

(n, {⋃_{i = 1}^{ℓ} P_{k_{i}}, K_{k + 1}})

and SPEX

(n, {⋃_{i = 1}^{ℓ} P_{k_{i}}, F_{k}})

for sufficiently large

n

, where

F_{k}

is a friendship graph on

2 k + 1

vertices consisting of

k

triangles which intersect in exactly one common vertex.

对于一组图F，如果图G不包含F中的任何图作为子图，则称为F自由图。设SPEX（n，F）表示在所有n阶的F-free图中具有最大谱半径的图。在本文中，对于任意非二部图H，我们给出了对于足够大的n，其中，r≥1,k≥1,k≥2，且存在至少一个ki不等于3的SPEX（n,{∈i=1∑Pki,H}）中的图的一些刻画。作为一个应用，对于足够大的n，我们完全刻画了SPEX（n,{∈i=1 l Pki,Kk+1}）和SPEX（n,{∈i=1 l Pki,Fk}）中的图，其中Fk是由k个恰好相交于一个公共顶点的k个三角形组成的2k+1个顶点的友谊图。

引用次数: 0

On partitions of edge-colored graphs under color degree constraints 色度约束下边色图的划分

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.dam.2025.12.031

Jing Lin , Huawen Ma

<div><div>A celebrated result of Stiebitz asserts that for positive integers <math><mi>s</mi></math> and <math><mi>t</mi></math>, each graph <math><mi>G</mi></math> with minimum degree <math><mrow><mi>s</mi><mo>+</mo><mi>t</mi><mo>+</mo><mn>1</mn></mrow></math> can be partitioned into vertex disjoint subgraphs <math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math> and <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math> such that <math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math> has minimum degree at least <math><mi>s</mi></math> and <math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math> has minimum degree at least <math><mi>t</mi></math>. Fujita et al. (2019) conjectured that the partition of Stiebitz can be extended to edge-colored graphs. Let <math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></mrow></math> be a graph, where <math><mi>c</mi></math> is an edge coloring of <math><mi>G</mi></math>. For a vertex <math><mi>v</mi></math> of <math><mi>G</mi></math>, let <math><mrow><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math> denote the edges of <math><mi>G</mi></math> incident to <math><mi>v</mi></math>, let <math><mrow><mi>d</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math> be the degree of <math><mi>v</mi></math> in <math><mi>G</mi></math>, let <math><mrow><mi>c</mi><mrow><mo>(</mo><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math> be the color set which contains all colors appearing on <math><mrow><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math> and let <math><mrow><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>c</mi><mrow><mo>(</mo><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math> be the color degree of <math><mi>v</mi></math> in <math><mi>G</mi></math>. Furthermore, let <math><mrow><msup><mrow><mi>δ</mi></mrow><mrow><mi>c</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mo>min</mo><mrow><mo>{</mo><msup><mrow><mi>d</mi></mrow><mrow><mi>c</mi></mrow></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>∣</mo><mi>v</mi><mo>∈</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math> be the minimum color degree of <math><mi>G</mi></math> (with respect to <math><mi>c</mi></math><

Stiebitz的一个著名结果断言，对于正整数s和t，每个最小度为s+t+1的图G都可以划分为顶点不相交子图G1和G2，使得G1的最小度至少为s， G2的最小度至少为t。Fujita et al.（2019）推测Stiebitz的划分可以推广到边色图。让G = (V, E、c)是一个图表,其中c是一个边缘着色G的顶点V (G)让E (V)表示G事件V的边缘,让d (V) = | E (V) | G V的程度,让c (E (V))的颜色集包含所有颜色出现在E (V)和直流(V) = | c (E (V) |在G .此外,V的颜色程度让δc (G) =分钟{直流(V)∣V∈V (G)}是最低程度的颜色G(关于c)。当δc(G[S])≥S且δc(G[T])≥T时，V(G)的分割（S，T）是（S，T）可行的。其中G[U]表示顶点集U诱导出的G的子图。Fujita、Li和Wang推测，在δc(G)≥s+t+1和s≥t≥2的条件下，G具有（s,t）可行划分。本文证明了如果s≥t≥2，且对于每个v∈v (G)， G具有（s,t）可行分割，且满足以下三个条件之一：(1)2dc(v)−d(v)≥s+t+1。(2) dc(v)≥ks+t+1，对于每个v∈v (G)， G相对于c的每个色类最大度数不超过k。(3)dc(v)≥s+t+1，对于每个v∈v (G)， c是传递着色(即如果P=(u,v,w)是G中的路径，且c(uv)=c(vw)，则uw∈E(G), c（uv)=c(vw)）。

引用次数: 0

The strong (2,2)-Conjecture for more classes of graphs 多类图的强（2,2）猜想

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.dam.2025.12.011

Olivier Baudon , Julien Bensmail , Morgan Boivin , Igor Grzelec , Clara Marcille

The Strong

(2, 2)

-Conjecture asks whether, for all connected graphs different from

K_{2}

and

K_{3}

, we can assign to edges red and blue labels with value 1 or 2 so that no two adjacent vertices have the same sum of incident red labels or the same sum of incident blue labels. This conjecture, which can be perceived as a generalisation of the so-called 1–2–3 Conjecture, as, thus far, been proved only for a handful number of graph classes. In this work, we prove the Strong

(2, 2)

-Conjecture holds for more classes of graphs. In particular, we prove the conjecture for cacti, subcubic outerplanar graphs, graphs with maximum average degree less than

\frac{9}{4}

, and some Halin graphs, among others.

Strong(2,2)-猜想问的是，对于所有不同于K2和K3的连通图，我们是否可以给边分配值为1或2的红色和蓝色标签，从而没有两个相邻的顶点具有相同的事件红色标签和相同的事件蓝色标签和。这个猜想，可以看作是所谓的1-2-3猜想的推广，到目前为止，只证明了少数图类。在这项工作中，我们证明了强(2,2)-猜想对更多的图类成立。特别地，我们证明了仙人掌图、次立方外平面图、最大平均度小于94的图和一些Halin图等的猜想。

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引用次数: 0

A note on valid inequalities for PageRank optimization with edge selection constraints 边选择约束下PageRank优化的有效不等式注释

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.dam.2025.12.023

Shang-Ru Yang, Yung-Han Liao, Chih-Ching Chien, Hao-Hsiang Wu

Csáji, Jungers, and Blondel prove that while a PageRank optimization problem with edge selection constraints is NP-hard, it can be solved optimally in polynomial time for the unconstrained case. This theoretical result is accompanied by several observations, which we leverage to develop valid inequalities in polynomial time for this class of NP-hard problems. We show that these observations can be exploited to derive stronger inequalities than the standard valid inequality available in the literature. These valid inequalities provide a theoretical basis for reducing the optimality gap of the constrained PageRank problem without changing NP-hardness.

Csáji、Jungers和Blondel证明，虽然具有边选择约束的PageRank优化问题是np困难的，但对于无约束情况，它可以在多项式时间内得到最优解。这个理论结果伴随着几个观察结果，我们利用这些观察结果在多项式时间内为这类np困难问题开发了有效的不等式。我们表明，这些观察可以被利用来推导出比文献中可用的标准有效不等式更强的不等式。这些有效不等式为在不改变np -硬度的情况下减小受限PageRank问题的最优性差距提供了理论基础。

引用次数: 0

Non-adaptive prophet inequalities for minor-closed classes of matroids 拟阵小闭类的非自适应先知不等式

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-15 DOI: 10.1016/j.dam.2025.12.001

Kanstantsin Pashkovich, Alice Sayutina

We consider the matroid prophet inequality problem. This problem has been extensively studied in the case of adaptive mechanisms. In particular, there is a tight 2-competitive mechanism for all matroids (Kleinberg and Weinberg, 2012).

However, it is not known what classes of matroids admit non-adaptive mechanisms with constant guarantee. Recently, in Chawla et al. (2024) it was shown that there are constant-competitive non-adaptive mechanisms for graphic matroids. In this work, we show that various known classes of matroids admit constant-competitive non-adaptive mechanisms.

考虑一类拟阵的先知不等式问题。这个问题在自适应机制中得到了广泛的研究。特别是，所有拟阵都存在紧密的2-竞争机制（Kleinberg和Weinberg， 2012）。然而，尚不清楚哪类拟阵具有恒定保证的非自适应机制。最近，Chawla等人（2024）表明，图形拟阵存在持续竞争的非自适应机制。在这项工作中，我们证明了各种已知类拟阵承认不断竞争的非适应机制。

引用次数: 0

Lines on digraphs of low diameter 低直径有向图上的线

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-13 DOI: 10.1016/j.dam.2025.12.015

Gabriela Araujo-Pardo , Martín Matamala , Juan Pablo Peña , José Zamora

A set of

n

non-collinear points in the Euclidean plane defines at least

n

different lines. Chen and Chvátal in 2008 conjectured that the same result is true in metric spaces for an adequate definition of line. More recently, it was conjectured in 2018 by Aboulker et al. that any large enough bridgeless graph on

n

vertices defines a metric space that has at least

n

lines.

We study the natural extension of Aboulker et al.’s conjecture into the context of quasi-metric spaces defined by digraphs of low diameter. We prove that it is valid for quasi-metric spaces defined by bipartite digraphs of diameter at most three, oriented graphs of diameter two and, digraphs of diameter three and directed girth four.

欧几里得平面上n个非共线点的集合定义了至少n条不同的直线。Chen和Chvátal在2008年推测，同样的结果在度量空间中也成立，因为有足够的线的定义。最近，Aboulker等人在2018年推测，任何足够大的n个顶点的无桥图都定义了一个至少有n条线的度量空间。我们研究了Aboulker等人的猜想在由低直径有向图定义的准度量空间中的自然推广。我们证明了它对于由直径最多为3的二部有向图、直径为2的有向图、直径为3的有向图和有向周长为4的有向图所定义的拟度量空间是有效的。

引用次数: 0

Bip-ordered bipartite Ramsey number 双序二部拉姆齐数

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2025-12-12 DOI: 10.1016/j.dam.2025.12.010

Ayun Zhang , Baoleer , Shinya Fujita , Yaping Mao , Gang Yang

<div><div>An ordered graph is a pair <math><mrow><msup><mrow><mi>G</mi></mrow><mrow><mo>≺</mo></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mo>≺</mo><mo>)</mo></mrow></mrow></math>, where <math><mo>≺</mo></math> is a total ordering of the vertex set <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. In this paper, we define a bip-ordered bipartite graph as a pair <math><mrow><msup><mrow><mi>G</mi></mrow><mrow><mo><</mo></mrow></msup><mo>=</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mo><</mo><mo>)</mo></mrow></mrow></math>, where <math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math> is a bipartite graph and where <math><mo><</mo></math> is a bip-order (for bipartite-order), namely, <math><mo><</mo></math> is a total bip-order on <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>X</mi><mo>∪</mo><mi>Y</mi></mrow></math> such that every vertex of <math><mi>X</mi></math> precedes every vertex of <math><mi>Y</mi></math> (that is, <math><mrow><mi>x</mi><mo><</mo><mi>y</mi></mrow></math>, for any <math><mrow><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><mi>X</mi><mo>×</mo><mi>Y</mi></mrow></math>). For bip-ordered bipartite graphs <math><mrow><msubsup><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo><</mo></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo><</mo></mrow></msubsup></mrow></math>, the bip-ordered bipartite Ramsey number <math><mrow><mover><mrow><mi>b</mi><mi>b</mi><mi>r</mi></mrow><mo>¯</mo></mover><mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow><mrow><mo><</mo></mrow></msubsup><mo>,</mo><mo>…</mo><mo>,</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mo><</mo></mrow></msubsup><mo>)</mo></mrow></mrow></math> is the least positive integer <math><mi>N</mi></math> for which the following property holds: for any <math><mi>k</mi></math>-edge-coloring of <math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>N</mi></mrow><mrow><mo><</mo></mrow></msubsup></math>, there is a color <math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math>, such that the subgraph of <math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>N</mi><mo>,</mo><mi>N</mi></mrow><mrow><mo><</mo></mrow></msubsup></math> induced by edges of color <math><mi>i</mi></math> contains a copy of <math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow><mrow><mo><</mo></mrow></msubsup></math></span

有序图是一对G =(G, )，其中的是顶点集合V(G)的总有序。本文将双序二部图定义为一对G<；=(G,<)，其中G=（X，Y，E）是二部图，且<；是双序（对于双序），即<；是V(G)=X∪Y上的全双序，使得X的每个顶点都先于Y的每个顶点（即X <； Y，对于任意（X，Y）∈X×Y）。对于双序二部图G1<；，…，Gk<，双序二部拉姆齐数bbr¯（G1<，…，Gk<）是满足以下性质的最小正整数N：对于KN，N<；的任意k边着色，存在一个颜色i∈{1，…，k}，使得由颜色i的边诱导的KN，N<；的子图包含一个Gi<；的副本（不一定是诱导的）。我们引入了当k=2时具有特定双序的一些基本双序图的双序二部拉姆齐数的界和精确值，以及当k≥2时基于Lovász局部引理的一般界。

引用次数: 0

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