Discrete Mathematics最新文献

英文中文

Improved upper bounds on Zarankiewicz numbers 改进了Zarankiewicz数的上界

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-11 DOI: 10.1016/j.disc.2025.114924

Sara Davies , Peter Gill , Daniel Horsley

For positive integers

s, t, m

and n, the Zarankiewicz number

z (m, n; s, t)

is the maximum number of edges in a subgraph of

K_{m, n}

that has no complete bipartite subgraph containing s vertices in the part of size m and t vertices in the part of size n. The best general upper bound on Zarankiewicz numbers is a bound due to Roman that can be viewed as the optimal value of a simple linear program. Here we show that in many cases this bound can be improved by adding additional constraints to this linear program. This allows us to prove new upper bounds on Zarankiewicz numbers for many small parameter sets. We are also able to establish a new family of closed form upper bounds on

z (m, n; s, t)

that captures much, but not all, of the power of the new constraints. This bound generalises a recent result of Chen, Horsley and Mammoliti that applied only in the case

s = 2

对于正整数s，t，m和n， Zarankiewicz数z（m,n;s,t）是在Km，n的子图中不存在大小为m的部分包含s个顶点和大小为n的部分包含t个顶点的完全二部子图的最大边数。Zarankiewicz数的最佳一般上界是由于Roman的一个界，可以看作是一个简单线性规划的最优值。在这里，我们证明了在许多情况下，这个边界可以通过给这个线性规划添加额外的约束来改进。这允许我们证明许多小参数集的Zarankiewicz数的新上界。我们还可以建立z（m,n;s,t）的一个新的封闭上界，它包含了新约束的大部分，但不是全部。这个界推广了Chen， Horsley和Mammoliti最近的一个结果，这个结果只适用于s=2的情况。

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引用次数: 0

A neighborhood union condition for the existence of a spanning tree without degree 2 vertices 无2次顶点的生成树存在的邻域联合条件

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-11 DOI: 10.1016/j.disc.2025.114944

Yibo Li , Fengming Dong , Xiaolan Hu , Huiqing Liu

For a connected graph G, a spanning tree T of G is called a homeomorphically irreducible spanning tree (HIST) if T has no vertices of degree 2. In this paper, we show that if G is a graph of order

n \geq 270

and

| N (u) \cup N (v) | \geq \frac{n - 1}{2}

holds for every pair of non-adjacent vertices u and v in G, then G has a HIST, unless G belongs to three exceptional families of graphs or G has a cut-vertex of degree 2. This result improves the latest conclusion, due to Ito and Tsuchiya, that the existence of a HIST in G can be guaranteed if

d (u) + d (v) \geq n - 1

holds for every pair of non-adjacent vertices u and v in G.

对于连通图G，如果T没有2次顶点，则G的生成树T称为同胚不可约生成树（HIST）。在本文中，我们证明了如果G是一个阶n≥270且| n (u)∪n (v)|≥n−12的图，对于G中每一对不相邻的顶点u和v都成立，那么G有一个HIST，除非G属于三个例外的图族或G有一个2次的切顶点。该结果改进了Ito和Tsuchiya的最新结论，即对于G中的每一对非相邻顶点u和v，如果d(u)+d(v)≥n−1成立，则G中存在HIST。

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引用次数: 0

On the spectral radius of unbalanced signed bipartite graphs 不平衡符号二部图的谱半径

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-11 DOI: 10.1016/j.disc.2025.114942

Cristian M. Conde , Ezequiel Dratman , Luciano N. Grippo

A signed graph is one that features two types of edges: positive and negative. Balanced signed graphs are those in which all cycles contain an even number of negative edges. In the adjacency matrix of a signed graph, entries can be 0, −1, or 1, depending on whether ij represents no edge, a negative edge, or a positive edge, respectively. The index of the adjacency matrix of a signed graph

\dot{G}

is less or equal to the index of the adjacency matrix of its underlying graph G, i.e.,

λ_{1} (\dot{G}) \leq λ_{1} (G)

. Indeed, if

\dot{G}

is balanced, then

λ_{1} (\dot{G}) = λ_{1} (G)

. This inequality becomes strict when

\dot{G}

is an unbalanced signed graph. Recently, Brunetti and Stanić found the whole list of unbalanced signed graphs on n vertices with maximum (resp. minimum) spectral radius. To our knowledge, there has been little research on this problem when unbalanced signed graphs are confined to specific graph classes. In this article, we demonstrate that there is only one unbalanced signed bipartite graph on n vertices with maximum spectral radius, up to an operation on the signed edges known as switching. Additionally, we investigate unbalanced signed complete bipartite graphs on n vertices with a bounded number of edges and maximum spectral radius, where the negative edges induce a tree.

带符号的图有两种边：正边和负边。平衡符号图是指所有环都包含偶数个负边的图。在有符号图的邻接矩阵中，根据ij代表的是无边、负边还是正边，表项可以是0、- 1或1。有符号图G˙的邻接矩阵索引小于等于其底层图G的邻接矩阵索引，即λ1（G˙）≤λ1(G)。实际上，如果G˙平衡，则λ1（G˙）=λ1(G)。当G˙是一个不平衡符号图时，这个不等式变得严格。最近，Brunetti和staniki发现了n个顶点上的最大不平衡符号图列表。最小)谱半径。据我们所知，当不平衡符号图局限于特定的图类时，对这一问题的研究很少。在这篇文章中，我们证明了在n个顶点上只有一个最大谱半径的不平衡有符号二部图，直到在有符号边上进行称为切换的操作。此外，我们研究了n个顶点上的非平衡有符号完全二部图，这些图具有有限的边数和最大的谱半径，其中负边诱导出树。

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引用次数: 0

Ramsey numbers of trees versus generalized wheels 树的拉姆齐数与广义轮

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-09 DOI: 10.1016/j.disc.2025.114938

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_{s, m}

denote a generalized wheel, obtained by joining each vertex of the complete graph

K_{s}

to every vertex of the cycle

C_{m}

. For

s \geq 2

m \geq 2

, and sufficiently large n, Chng, Tan, and Wong (Discrete Math., 2021) conjectured that

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

In this note, we confirm this conjecture in a stronger form by providing a linear lower bound on n in terms of m for which the equality holds.

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。设Tn表示有n个顶点的树，设Ws，m表示一个广义轮，通过将完全图k的每个顶点与循环Cm的每个顶点连接而得到。对于s≥2，m≥2，和足够大的n， cheng, Tan, and Wong（离散数学）。, 2021)推测thatR (Tn Ws 2米)= (s + 1) (n−1)+ 1。在这篇笔记中，我们以一种更强的形式证实了这个猜想，我们用m给出了n的线性下界，在这个下界中等式成立。

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引用次数: 0

Recursion polynomial for cubic rotation symmetric Boolean functions 三次旋转对称布尔函数的递归多项式

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-09 DOI: 10.1016/j.disc.2025.114912

Thomas W. Cusick , Younhwan Cheon

<div><div>Rotation symmetric (RS) Boolean functions have been extensively studied for over twenty years because of their applications in cryptography and coding theory. The present paper studies degree 3 RS functions, and relies extensively on the theory of the affine equivalence of such functions developed in [3]. It is known [1] that if <math><mi>f</mi><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> is the RS Boolean function in n variables generated by the monomial <math><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub></math> (notation <math><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math>), then the sequence <math><mi>w</mi><mi>t</mi><mo>(</mo><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo><mo>,</mo><mspace></mspace><mi>n</mi><mo>=</mo><mi>i</mi><mo>,</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo>…</mo></math>, where <math><mi>w</mi><mi>t</mi><mo>(</mo><msub><mrow><mo>(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>x</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> denotes the Hamming weight of the function, satisfies a linear recursion with integer coefficients and this recursion can be explicitly computed with a method given in [1]. It was observed in [10, Lemma 3.5, p. 396] that the functions <math><msub><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>4</mn><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math> and <math><msub><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mn>5</mn><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msub></math> have the same weights for every n even though the two functions are not affine equivalent for infinitely many values of n. It was not clear what the explanation for that is. This paper answers that question and gives a general theory that provides

旋转对称布尔函数由于在密码学和编码理论中的应用，在过去的二十多年里得到了广泛的研究。本文研究了3度RS函数，并广泛依赖于[3]中该类函数的仿射等价理论。我们知道，如果f（x1,x2，…，xn）是由单项式x1，x2，…xi（符号（x1,x2，…，xi）n）生成的n个变量中的RS布尔函数，则序列wt((x1,x2，…，xi)n),n=i,i+1，…，其中wt（(x1,x2，…，xi)n）表示函数的汉明权，满足整数系数线性递归，该递归可以用[1]中给出的方法显式计算。在[10，引理3.5,p. 396]中观察到，函数（1,2,4）n和（1,2,5）n对于每个n具有相同的权值，尽管这两个函数对于无限多个n值不是仿射等价的。对此的解释并不清楚。本文回答了这个问题，并给出了一个一般理论，该理论为函数对（1,r,s）n和（1,t,u）n提供了更多类似行为的例子。

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 : 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

Trees with one as Laplacian eigenvalue with multiplicity two less than the number of pendant vertices 以拉普拉斯特征值为1的树，其多重度小于垂顶点数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-09 DOI: 10.1016/j.disc.2025.114922

Vinayak Gupta

Let T be a tree with p pendant vertices, and let

m (T, λ)

denote the multiplicity of the eigenvalue λ of the Laplacian matrix (T). It has recently been shown that

m (T, 1) = p - 1

if and only if T has p pendant vertices and the distance between any two distinct pendant vertices u and v satisfies

d (u, v) \equiv 2 \mod 3

. This article provides a complete characterization of all trees T for which

m (T, 1) = p - 2

设T是一棵有p个垂顶点的树，设m（T,λ）表示拉普拉斯矩阵(T)的特征值λ的多重性。最近已经证明m(T,1)=p−1当且仅当T有p个垂顶点，且任意两个不同垂顶点u和v之间的距离满足d(u,v)≡2mod3。本文给出了m(T,1)=p−2的所有树T的完整刻画。

引用次数: 0

Independent sets in K6-free graphs with large degree 大度k6自由图中的独立集

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-09 DOI: 10.1016/j.disc.2025.114941

Jeremy Lyle

In [2], Brandt conjectured that

K_{r}

-free graphs on n vertices with minimum degree larger than

((5 r - 11) / (5 r - 5)) n

have independent sets on at least

(1 / (r - 1)) n

vertices. This result has been shown for

r = 3, 4, 5

, but not for larger r. In this paper, we provide a simple result that the conjecture holds for

r = 6

as well; that is, any

K_{6}

-free graph with minimum degree larger than

(19 / 25) n

has an independent set of order at least

(1 / 5) n

在[2]中，Brandt推测最小度大于((5r−11)/(5r−5))n的n个顶点上的无k图在至少（1/(r−1)）n个顶点上具有独立集。这一结果在r=3,4,5时已被证明，但在更大的r时则未被证明。在本文中，我们提供了一个简单的结果，即该猜想在r=6时也成立；即任何最小度大于（19/25）n的无k6图都有一个阶数至少为（1/5）n的独立集。

引用次数: 0

Minimal linear codes from vectorial functions 向量函数的最小线性码

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-09 DOI: 10.1016/j.disc.2025.114940

Yanjun Li , Haibin Kan , Fangfang Liu , Jie Peng , Lijing Zheng , Zepeng Zhuo

The study on minimal linear codes has received great attention, due to their significant applications in secret sharing schemes and secure two-party computation. Until now, numerous minimal linear codes have been discovered. However, only a few infinite families of minimal linear codes were found from vectorial functions. In this paper, we present a necessary and sufficient condition such that a large class of ternary linear codes from vectorial functions is minimal and violates the AB condition simultaneously, by using the Walsh transform of vectorial functions. We also give a sufficient condition such that a large family of linear codes from vectorial functions is minimal, by using cutting blocking sets. These two results extend two main results of [22] and [7] to the case of vectorial functions, respectively. According to these two results, we find several minimal linear codes violating the AB condition, respectively.

由于极小线性码在秘密共享方案和安全计算中的重要应用，其研究受到了广泛的关注。到目前为止，已经发现了许多最小线性码。然而，从向量函数中只发现了少数无限族的最小线性码。本文利用向量函数的Walsh变换，给出了一类由向量函数构成的三元线性码极小且同时违反AB条件的充要条件。利用切块集，给出了向量函数的一大族线性码是极小的充分条件。这两个结果分别将[22]和[7]的两个主要结果推广到向量函数的情况。根据这两个结果，我们分别找到了几个违背AB条件的最小线性码。

引用次数: 0

A spectral stability result regarding the complete bipartite graph K2,t 关于完全二部图K2，t的谱稳定性结果

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2025-12-08 DOI: 10.1016/j.disc.2025.114914

Ruike Wang, Zhenzhen Lou

Spectral stability theorems have been a crucial aspect of graph theory research. Consider a graph G with size m and spectral radius

ρ (G)

. Building on the solid foundation laid by previous works in this rich field, this paper presents novel and valuable findings related to stability. Wang and Guo (2024) [16] showed an important result. Given that

m = Ω (k^{4})

and

k \geq 0

, when

ρ (G) \geq \sqrt{m - k}

, then G contains either a quadrilateral or a star of size

m - k

. In this paper, we take a significant step forward by generalizing this result. Precisely, for

m = Ω (k^{4})

and

2 \leq t \leq k + 2

, when

ρ (G) \geq \sqrt{m - k + t - 2}

, we prove that G contains either a copy of

K_{2, t}

(a complete bipartite graph with two vertices on one side and t vertices on the other side) or a star of size

m - k + t - 2

. This generalization contributes to a more profound understanding of the spectral and structural aspects of graphs, as well as their stability properties.

谱稳定性定理一直是图论研究的一个重要方面。考虑一个大小为m，谱半径为ρ(G)的图G。本文在前人在这一丰富领域的工作奠定的坚实基础上，提出了与稳定性有关的新颖而有价值的发现。Wang and Guo(2024)[16]给出了重要的结果。设m=Ω（k4）且k≥0，当ρ(G)≥m - k时，则G包含大小为m - k的四边形或星形。在本文中，我们通过推广这一结果向前迈出了重要的一步。准确地说，对于m=Ω（k4）和2≤t≤k+2，当ρ(G)≥m−k+t−2时，我们证明了G包含K2的一个副本，t（一侧有两个顶点，另一侧有t个顶点的完全二部图）或一个大小为m−k+t−2的星。这种推广有助于更深刻地理解图的谱和结构方面，以及它们的稳定性。

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