Discrete Mathematics最新文献

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Trisimplicial vertices in (fork, odd parachute)-free graphs 无（叉，奇降落伞）图中的三单纯顶点

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-04 DOI: 10.1016/j.disc.2025.114920

Kaiyang Lan , Feng Liu , Di Wu , Yidong Zhou

An odd hole in a graph is an induced subgraph which is a cycle of odd length at least five. An odd parachute is a graph obtained from an odd hole H by adding a new edge uv such that x is adjacent to u but not to v for each

x \in V (H)

. A graph G is perfectly divisible if for each induced subgraph H of G,

V (H)

can be partitioned into A and B such that

H [A]

is perfect and

ω (H [B]) < ω (H)

. A vertex of a graph is trisimplicial if its neighborhood is the union of three cliques. In this paper, we prove that if G is a (fork, odd parachute)-free graph, then G is either perfectly divisible or has a trisimplicial vertex, from which we deduce that every nonperfectly divisible claw-free graph contains a trisimplicial vertex. As an application, we show that

χ (G) \leq (\begin{matrix} ω (G) + 1 \\ 2 \end{matrix})

if G is a (fork, odd parachute)-free graph. This generalizes some results of Karthick et al. (2022) [11], and Wu and Xu (2024) [20].

图中的奇孔是一个诱导子图，它是一个奇长度至少为5的循环。奇伞是通过添加一条新边uv从奇洞H得到的图，使得对于每个x∈v (H)， x与u相邻，但不与v相邻。如果对于G的每个诱导子图H， V(H)可以划分为A和B，使得H[A]是完全的，ω(H[B])<ω(H)，则图G是完全可分的。如果一个图的顶点的邻域是三个团的并集，那么它就是三单纯的。本文证明了如果G是（叉，奇降落伞）自由图，则G要么完全可分，要么有一个三分顶点，由此推导出每一个不可完全可分的无爪图都包含一个三分顶点。作为一个应用，我们证明了χ(G)≤(ω(G)+12)，如果G是一个（叉，奇降落伞）自由图。这概括了Karthick et al.(2022)[11]和Wu and Xu(2024)[20]的一些结果。

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引用次数: 0

Identical representation functions of linear forms. I 线性形式的相同表示函数。我

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-03 DOI: 10.1016/j.disc.2025.114910

Sándor Z. Kiss , Csaba Sándor

For a set of natural numbers A, let

R_{A} (n)

be the number of representations of a natural number n as the sum of two terms from A. Many years ago, Nathanson studied the conditions for the sets A and B of natural numbers that are needed to guarantee that

R_{A} (n) = R_{B} (n)

for every positive integer n. In the last decades, similar questions have been studied by many scholars. In this paper, we extend Nathanson's result to representation functions associated to linear forms and we study related problems.

对于自然数集合a，设RA(n)为自然数n的两项之和的表示个数。多年前，Nathanson研究了自然数集合a和集合B保证RA(n)=RB(n)对每一个正整数n的条件。近几十年来，许多学者研究了类似的问题。本文将Nathanson的结果推广到与线性形式相关的表示函数，并研究了相关问题。

引用次数: 0

On subsequence sums of index-1-free sequences over cyclic groups 关于循环群上无索引序列的子序列和

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-03 DOI: 10.1016/j.disc.2025.114918

Jiangtao Peng , Shijie Yuan , Yuanlin Li

Let G be a cyclic group of order n. Every finite sequence S of elements from G can be written in the form

S = (x_{1} g) \cdot \dots \cdot (x_{ℓ} g)

, where

g \in G

with

〈 g 〉 = G

and

x_{1}, \dots, x_{ℓ} \in [1, n]

. The index of S is defined to be the minimum of

(x_{1} + \dots + x_{ℓ}) / n

over all possible generator

g \in G

. We call S index-1-free, if S contains no subsequence of index 1. Gao conjectured that if S is an index-1-free sequence, then S has at least

| S |

distinct subsequence sums, where the subsequences are of index less than 1. In this paper, we confirm the conjecture for certain cases, and also provide counterexamples to the conjecture.

设G是一个n阶的循环群，由G上的元素组成的有限序列S可以写成S=(x1g)⋅…⋅(x∑G)，其中G∈G， < G > =G，且x1，…，x∑∈[1,n]。S的指标被定义为（x1+…+x l）/n在所有可能的生成子g∈g上的最小值。如果S不包含索引1的子序列，我们称S为索引1 free。Gao推测，如果S是一个索引不为1的序列，则S至少有|S|个不同的子序列和，其中子序列的索引小于1。本文在某些情况下证实了这一猜想，并给出了反例。

引用次数: 0

Secure domination in P5-free graphs P5-free图中的安全支配

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-02 DOI: 10.1016/j.disc.2025.114905

Uttam K. Gupta , Michael A. Henning , Paras Vinubhai Maniya , Dinabandhu Pradhan

A dominating set of a graph G is a set

S \subseteq V (G)

such that every vertex in

V (G) ∖ S

has a neighbor in S, where two vertices are neighbors if they are adjacent. A secure dominating set of G is a dominating set S of G with the additional property that for every vertex

v \in V (G) ∖ S

, there exists a neighbor u of v in S such that

(S ∖ {u}) \cup {v}

is a dominating set of G. The secure domination number of G, denoted by

γ_{s} (G)

, is the minimum cardinality of a secure dominating set of G. We prove that if G is a

P_{5}

-free graph, then

γ_{s} (G) \leq \frac{3}{2} α (G)

, where

α (G)

denotes the independence number of G. We further show that if G is a connected

(P_{5}, H)

-free graph for some

H \in {P_{3} \cup P_{1}, K_{2} \cup 2 K_{1}, paw, C_{4}}

, then

γ_{s} (G) \leq \max {3, α (G)}

. We also show that if G is a

(P_{3} \cup P_{2})

-free graph, then

γ_{s} (G) \leq α (G) + 1

图G的支配集是一个集S⊥V(G)，满足V(G)∑S中的每个顶点在S中有一个邻居，其中两个顶点相邻为邻居。G的安全控制集是G的控制集S，其附加性质是对于每个顶点v∈v (G)∑S，在S中存在一个v的邻居u，使得(S∈{u})∪{v}是G的控制集。G的安全控制数，记作γs(G)，是G的安全控制集的最小cardinality。我们证明如果G是P5-free图，则γs(G)≤32α(G)，其中α(G)表示G的独立数，进一步证明了对于某些H∈{P3∪P1，K2∪2K1,paw,C4}，如果G是连通（P5，H）自由图，则γs(G)≤max (3,α(G)}。我们还证明了如果G是一个（P3∪P2）自由图，那么γs(G)≤α(G)+1。

引用次数: 0

On representation functions related to the partition 论与分区相关的表示函数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-02 DOI: 10.1016/j.disc.2025.114908

Fang-Gang Xue , Xue-Qin Cao

<div><div>Let <math><mi>Z</mi></math> be the set of integers and <math><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></math> (resp. <math><mi>N</mi></math>) the set of non-negative (resp. positive) integers. For a nonempty set of integers A and integers n, <math><mi>h</mi><mo>≥</mo><mn>2</mn></math>, denote <math><msub><mrow><mi>r</mi></mrow><mrow><mi>A</mi><mo>,</mo><mi>h</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> by the number of representations of n of the form <math><mi>n</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></math>, where <math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≤</mo><mo>⋯</mo><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>h</mi></mrow></msub></math> and <math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>∈</mo><mi>A</mi></math> for <math><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>h</mi></math>, and <math><msub><mrow><mi>d</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math> by the number of <math><mo>(</mo><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></math> with <math><mi>a</mi><mo>,</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>∈</mo><mi>A</mi></math> such that <math><mi>n</mi><mo>=</mo><mi>a</mi><mo>−</mo><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup></math>. Following Lev's work, we prove that there is a partition <math><mi>Z</mi><mo>=</mo><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub></math> of the set of all integers such that, for each <math><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub></math>, we have <math><msub><mrow><mi>r</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><mn>2</mn></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math> for all integers n and <math><msub><mrow><mi>d</mi></mrow><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mn>1</mn></math> for all positive integers n. As a main result, we also prove that, for any integers <math><mi>h</mi><mo>≥</mo><mn>2</mn></math> and <math><mi>m</mi><mo>≥</mo><mn>2</mn></math>, there exists a set T of integers with the density <math><mn>1</mn><mo>−</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>m</mi></mrow></mfrac></mat

设Z为整数的集合，N0 (p。N)非负的(resp。积极的)整数。对于整数a和整数n， h≥2的非空集合，用n的表示形式n=a1+a2+⋯+ah的个数表示rA，h(n)，其中a1≤，≤ah，且对于i=1,2,⋯,h, dA(n)用（a,a’）的个数表示，其中a，a’∈a使得n=a - a’。根据Lev的工作，我们证明了所有整数集合的一个分区Z= k=1∞Ak，使得对于每一个Ak，我们有rAk，2(n)=1，对于所有正整数n，我们有dAk(n)=1。作为一个主要结果，我们还证明了对于任何整数h≥2，m≥2，存在一个密度为1 - 1m的整数集合T，对于任何函数f:Z→N0′{∞}，f−1(0)=T，存在一个整数集合a，满足rA，h(n)=f(n)对于所有n∈Z。这提供了一个与内特森问题相关的特定密度结果。

引用次数: 0

Binary trees with extremal number of maximal independent sets 具有极大独立集的极数的二叉树

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-02 DOI: 10.1016/j.disc.2025.114911

Opeyemi Oyewumi , Adriana Roux , Stephan Wagner

A binary tree (more precisely, an unrooted binary tree) is a tree in which all internal vertices (i.e., non-leaves) are exactly of degree 3. We give an upper bound and a lower bound for the number of maximal independent sets in binary trees together with a characterization of the extremal binary trees. The binary trees with second largest number of maximal independent sets are also characterized.

二叉树（更准确地说，是一棵无根二叉树）是一棵所有内部顶点（即非叶子）都恰好是3度的树。给出了二叉树中最大独立集个数的上界和下界，并给出了极值二叉树的特征。对最大独立集个数第二多的二叉树也进行了刻画。

引用次数: 0

An improved upper bound on the covering radius of the logarithmic lattice of Q(ζn) Q（ζn）对数格覆盖半径的改进上界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-01 DOI: 10.1016/j.disc.2025.114909

James Punch

Let

R^{m}

be endowed with the Euclidean metric. The covering radius of a lattice

Λ \subset R^{m}

is the least distance r such that, given any point of

R^{m}

, the distance from that point to Λ is not more than r. Lattices can occur via the unit group of the ring of integers in an algebraic number field

K

, by applying a logarithmic embedding

K^{⁎} \to R^{m}

. In this paper, we examine those lattices which arise from the cyclotomic number field

Q (ζ_{n})

, for a given positive integer

n \geq 5

such that

n ≢ 2 (\mod 4)

. We then provide improvements to a result of de Araujo in [3], and conclude with an upper bound on the covering radius for this lattice in terms of n and the number of its distinct prime factors. In particular, we improve [3, Lemma 2], and show that, asymptotically, it can be improved no further.

让Rm被赋予欧几里德度规。晶格Λ∧Rm的覆盖半径是最小距离r，使得给定Rm的任意一点，从该点到Λ的距离不大于r。通过应用对数嵌入K→Rm，可以通过代数数域K中的整数环的单位群出现晶格。在本文中，对于给定正整数n≥5，我们研究了由分环数域Q（ζn）产生的格，使得n 2（mod4）。然后，我们对[3]中de Araujo的结果进行了改进，并以n及其不同素数因子的个数给出了该格的覆盖半径的上界。特别地，我们改进了[3，引理2]，并证明，渐近地，它不能再改进了。

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引用次数: 0

(θ,δ)-Skew generalized quasi-cyclic codes over the ring R=Z4+uZ4 环R=Z4+uZ4上的(θ,δ)-偏广义拟循环码

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-01 DOI: 10.1016/j.disc.2025.114906

Ayoub Mounir, Abdelfattah Haily, Mohammed El Badry

In this paper, we consider the finite non-chain ring

R = Z_{4} + u Z_{4}

with

u^{2} = 1

. We provide a new class of codes, known as

(θ, δ)

-skew generalized quasi-cyclic (GQC) codes over R, where θ is an automorphism of R and δ is a θ-derivation of R. This work generalizes

(θ, δ)

-skew quasi-cyclic (QC) codes. We give the structure of 1-generator

(θ, δ)

-skew GQC codes over R, and we provide a sufficient condition for 1-generator

(θ, δ)

-skew GQC code over R to be free. A lower bound of the minimum distance of free 1-generator

(θ, δ)

-skew GQC codes is also given. Moreover, we present some numerical examples in which we derive new

Z_{4}

-linear codes through the application of the Gray map. Furthermore, we characterize the Euclidean dual codes of

(θ, δ)

-skew GQC codes.

本文考虑了u2=1的有限非链环R=Z4+uZ4。本文提出了一类新的码，即R上的(θ,δ)-偏广义拟循环码（GQC），其中θ是R的自同构，δ是R的θ导数。给出了R上1-generator (θ,δ)-skew GQC码的结构，并给出了R上1-generator (θ,δ)-skew GQC码自由的充分条件。给出了自由1-发生器（θ,δ）偏态GQC码最小距离的下界。此外，我们还给出了一些数值例子，其中我们通过应用灰度图推导出新的z4 -线性码。此外，我们还刻画了（θ,δ）偏态GQC码的欧几里得对偶码。

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引用次数: 0

On forbidding graphs as traces of hypergraphs 关于禁止图作为超图的轨迹

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-01 DOI: 10.1016/j.disc.2025.114907

Dániel Gerbner , Michael E. Picollelli

We say that a hypergraph

H

contains a graph H as a trace if there exists some set

S \subset V (H)

such that

H |_{S} = {h \cap S : h \in E (H)}

contains a subhypergraph isomorphic to H. We study the largest number of hyperedges in 3-uniform hypergraphs avoiding some graph F as trace. In particular, we improve a bound given by Luo and Spiro in the case

F = C_{4}

, and obtain exact bounds for large n when F is a book graph.

如果存在某个集合S∧V(H)使得H bb 0 S={H∩S: H∈E(H)}包含与H同构的子超图H，则我们说超图H包含图H作为迹。我们研究3-一致超图中避免图F作为迹的最大超边数。特别地，我们改进了Luo和Spiro在F=C4情况下给出的界，得到了当F为book图时n大时的精确界。

引用次数: 0

Enumerating minimal dominating sets in the (in)comparability graphs of bounded dimension posets 列举有界维序集可比性图中的最小支配集

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-11-28 DOI: 10.1016/j.disc.2025.114904

Marthe Bonamy , Oscar Defrain , Piotr Micek , Lhouari Nourine

Enumerating minimal transversals in a hypergraph is a notoriously hard problem. It can be reduced to enumerating minimal dominating sets in a graph, in fact even to enumerating minimal dominating sets in an incomparability graph. We provide an output-polynomial time algorithm for incomparability graphs whose underlying posets have bounded dimension. Through a different proof technique, we also provide an output-polynomial time algorithm for their complements, i.e., for comparability graphs of bounded dimension posets. Our algorithm for incomparability graphs relies on the geometrical representation of incomparability graphs with bounded dimension, as given by Golumbic et al. in 1983. It runs with polynomial delay and only needs polynomial space. Our algorithm for comparability graphs is based on the flipping method introduced by Golovach et al. in 2015. It performs in incremental-polynomial time and possibly requires exponential space.

在超图中枚举最小截线是一个众所周知的难题。它可以简化为在图中枚举最小控制集，实际上甚至可以简化为在不可比较图中枚举最小控制集。我们提供了一个不可比较图的输出多项式时间算法，其底层偏序集有界维。通过一种不同的证明技术，我们也为它们的补提供了一个输出多项式时间算法，即对于有界维序集的可比性图。我们的不可比较图算法依赖于有界维不可比较图的几何表示，如Golumbic等人在1983年给出的。它以多项式延迟运行，只需要多项式空间。我们的可比性图算法基于Golovach等人在2015年引入的翻转方法。它在增量多项式时间内执行，并且可能需要指数空间。

引用次数: 0

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