Discrete Mathematics最新文献

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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 : 2026-04-01 Epub 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

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 : 2026-04-01 Epub 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

Sufficient conditions for edge-colored bipartite graphs to have rainbow and properly colored spanning trees 边色二部图具有彩虹和适当着色生成树的充分条件

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-04 DOI: 10.1016/j.disc.2025.114921

Shinya Fujita , Mikio Kano , Shun-ichi Maezawa , Masao Tsugaki , Kiyoshi Yoshimoto

An edge-colored graph is called rainbow if no two edges have the same color. An edge-colored graph is called properly colored if no two adjacent edges have the same color. We consider a connected edge-colored bipartite graph

G = G (A, B)

with bipartition

(A, B)

in which for every color c, the set of edges colored with c forms a star. It is known that if the minimum color degree of G is at least

| G | / 3 + 1

, then G has a rainbow spanning tree. In this paper, we prove that if the minimum color degree of G is at least

(\max {| A |, | B |} + 2) / 2

, then G has a rainbow spanning tree. Our theorem strengthens the previous result. We also provide a similar sufficient condition for connected edge-colored bipartite graphs to have properly colored spanning trees. Moreover, we show that these minimum color degree conditions are sharp.

如果没有两条边的颜色相同，则称为彩虹。如果没有两条相邻的边具有相同的颜色，则称为正确着色的图。我们考虑一个具有二分（a，B）的连通边着色二部图G=G(a，B)，其中对于每一种颜色c，用c着色的边的集合形成一个星形。已知，如果G的最小色度至少为|G|/3+1，则G具有彩虹生成树。在本文中，我们证明了如果G的最小色度至少为(max {|A|,|B|}+2)/2，则G存在彩虹生成树。我们的定理加强了前面的结果。我们也给出了连通边着色二部图具有适当着色生成树的一个类似的充分条件。此外，我们表明这些最小色度条件是清晰的。

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引用次数: 0

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 : 2026-04-01 Epub 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

On a conjecture of regular graphs having the minimum number of induced paths of length two 关于具有最小长度为2的诱导路径数的正则图的一个猜想

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-09 DOI: 10.1016/j.disc.2025.114936

Shi-Cai Gong , Ni Yang , Jia-Jin Wang , Ya-Hong Chen

Let

t (G)

be the number of spanning trees of a simple graph G, and let

Γ (n, m)

denote the class of all n-vertex m-edge simple graphs. A graph

G \in Γ (n, m)

is called t-optimal if

t (G) \geq t (H)

for every

H \in Γ (n, m)

. Petingi and Rodríguez (Discrete Math., 2002) proved that, for n larger than an explicit threshold

n_{0}

, any t-optimal graph has an almost-regular complement containing the minimum possible number of induced 2-edge paths. Furthermore, they proposed a conjecture regarding graphs having the minimum number of induced paths of length two within

Γ (n; m)

We confirm the conjecture above for all regularity degrees

k \in {1, 2, 3, 4, 5}

. As a by-product, the t-optimal members of

Γ (n, n (n - 5) / 2)

and

Γ (n, n (n - 6) / 2)

are completely determined for all

n > n_{0}

, where the threshold

n_{0}

can be explicitly determined.

设t(G)为简单图G的生成树个数，设Γ（n,m）为所有n顶点m边简单图的类。对于每个H∈Γ（n,m），如果t(G)≥t(H)，则图G∈Γ（n,m）称为t-最优。Petingi和Rodríguez（离散数学）。， 2002)证明了，当n大于显式阈值n0时，任何t-最优图都有一个包含诱导2边路径的最小可能数的几乎正则补。此外，他们提出了一个关于在Γ（n;m）内具有最小长度为2的诱导路径数的图的猜想。对于所有正则度k∈{1,2,3,4,5}，我们证实了上述猜想。作为副产品，对于所有n>；n0，可以完全确定Γ（n,n(n−5)/2）和Γ（n,n(n−6)/2）的t-最优成员，其中阈值n0可以显式确定。

引用次数: 0

The vertex-face chromatic number of almost all nonorientable surfaces 几乎所有不可定向曲面的顶点面色数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-17 DOI: 10.1016/j.disc.2025.114950

Vladimir P. Korzhik

The vertex-face chromatic number

χ_{v f} (N_{q})

of a nonorientable surface

N_{q}

of genus q is the minimum integer m such that the vertices and faces of any map on the surface can be colored by m colors in such a way that adjacent or incident elements receive distinct colors. The known upper bound

Φ (N_{q})

χ_{v f} (N_{q})

differs from Ringel's upper bound on the 1-chromatic number of nonorientable surfaces.

The results are as follows:

(i)
There is a constant $A > 0$ such that for any $q \geq A$ , $Φ (N_{q}) - 1 \leq χ_{v f} (N_{q}) \leq Φ (N_{q})$ .
$(i i)$
Let x be an integer and let $P (x)$ be the number of values of q in the interval $[1, x]$ such that $χ_{v f} (N_{q}) = Φ (N_{q})$ . Then $\lim_{x \to \infty} P (x) / x = 1$ .

不可定向曲面Nq的顶点面色数χvf（Nq）是最小整数m，使得曲面上任何地图的顶点和面都可以用m种颜色着色，从而使相邻或相关元素获得不同的颜色。已知的χvf（Nq）上界Φ（Nq）不同于不可定向曲面1色数上的Ringel上界。结果如下：(i)存在一个常数A>；0，使得对于任意q≥a， Φ(Nq)−1≤χvf(Nq)≤Φ（Nq）。（ii）设x为整数，设P(x)为区间[1，x]中使χvf(Nq)=Φ（Nq）的q值的个数。然后limx→∞⁡P (x) / x = 1。

引用次数: 0

New results in vertex sedentariness 顶点静坐的新结果

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-22 DOI: 10.1016/j.disc.2025.114959

Hermie Monterde

A vertex in a graph is said to be sedentary if a quantum state assigned to that vertex tends to stay on that vertex. Using the direct product, join, and blow-up operations, we construct new infinite families of graphs with sedentary vertices. We prove our main result, which states that a vertex with a twin is either sedentary or admits pretty good state transfer with its twin. In contrast, we provide infinite families of graphs containing vertices with no twins that are neither sedentary nor involved in pretty good state transfer. Using our main result, we completely characterize twin vertices that are sedentary and provide improved bounds on their sedentariness. This allows us to construct an infinite family of trees with maximum degree three that admit vertex sedentariness. We also determine all complete bipartite graphs and threshold graphs that contain sedentary vertices relative to the Laplacian matrix.

如果分配给一个顶点的量子态倾向于停留在该顶点上，则图中的顶点被称为不动顶点。利用直积、连接和爆破运算，我们构造了具有固定顶点的无限族图。我们证明了我们的主要结果，即具有双胞胎的顶点要么是静止的，要么可以很好地与其双胞胎进行状态转移。相比之下，我们提供了无限的图族，这些图族包含没有双胞胎的顶点，既不是静止的，也不是很好的状态转移。利用我们的主要结果，我们完全描述了久坐不动的双顶点，并提供了改进的久坐不动的边界。这允许我们构造一个无限的树族，最大度为3，允许顶点静坐。我们还确定了包含相对于拉普拉斯矩阵的不动顶点的所有完备二部图和阈值图。

引用次数: 0

On nonsingular weightwise perfectly balanced functions 在非奇异权完全平衡函数上

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-04-01 Epub Date: 2025-12-05 DOI: 10.1016/j.disc.2025.114919

Xinyi Zhao , Shizhu Tian , Xiangyong Zeng , Zhimin Sun

Boolean functions restricted to some subset of

F_{2}^{n}

play an important role in the FLIP stream cipher proposed by Méaux et al. in 2016. Carlet et al. later studied the particular case where the subset is composed of vectors with constant Hamming weight, known as weightwise perfectly balanced (WPB) functions. Motivated by the application of nonsingular functions in stream ciphers based on feedback shift registers (FSRs), in this paper, we focus on nonsingular WPB functions. Firstly, we propose an algorithm based on the truth tables to generate all nonsingular WPB functions. A concrete construction of such functions is also provided. Considering the security in different application scenarios, we further investigate their nonlinearity and cycle structure. A tight lower bound on the nonlinearity is obtained, as well as a theoretical probability distribution of the nonlinearity. Notably, our lower bound is higher than the known lower bound on WPB functions. To generate nonsingular WPB functions with high nonlinearity, an improved algorithm is also proposed. Finally, we offer a method to analyze the cycle structure of nonsingular WPB functions, and a method to obtain such functions which generate a few cycles. Experiments show that nonsingular WPB functions are suitable for constructing de Bruijn sequences with cycle-joining method.

限制于F2n的某个子集的布尔函数在msamuaux等人于2016年提出的FLIP流密码中发挥了重要作用。Carlet等人后来研究了一种特殊情况，其中子集由具有恒定汉明权重的向量组成，称为加权完美平衡（WPB）函数。基于非奇异函数在基于反馈移位寄存器（FSRs）的流密码中的应用，本文重点研究了非奇异WPB函数。首先，我们提出了一种基于真值表的非奇异WPB函数生成算法。还提供了这些函数的具体结构。考虑到不同应用场景下的安全性，我们进一步研究了它们的非线性和周期结构。得到了非线性的一个紧下界，以及非线性的理论概率分布。值得注意的是，我们的下界高于已知的WPB函数的下界。为了生成高度非线性的非奇异WPB函数，提出了一种改进算法。最后，我们给出了一种分析非奇异WPB函数的环结构的方法，以及一种获取产生少量环的非奇异WPB函数的方法。实验表明，非奇异WPB函数适用于用循环连接法构造de Bruijn序列。

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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 : 2026-04-01 Epub 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 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 : 2026-04-01 Epub 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

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