Discrete Mathematics最新文献

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On cyclic P(4n,2n − 1)-designs 循环P(4n,2n − 1)-设计

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-16 DOI: 10.1016/j.disc.2026.115000

Wannasiri Wannasit

We show that the generalized Petersen graphs

P (4 n, 2 n - 1)

admits a

ρ^{+}

-labeling for every positive integer n. In this way, we obtain the existence of a cyclic

P (4 n, 2 n - 1)

-decomposition of

K_{v}

for every

v \equiv 1 (\mod 24 n)

我们证明了广义Petersen图P（4n,2n−1）对于每一个正整数n都有一个ρ+标记。这样，我们得到了对于每一个v≡1(mod24n)， Kv存在一个循环P（4n,2n−1）分解。

引用次数: 0

Kempe equivalence of 4-colorings of graphs on non-orientable surfaces 非定向曲面上图的四色的Kempe等价

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-16 DOI: 10.1016/j.disc.2026.115007

Naoki Matsumoto

Two vertex colorings of a graph are Kempe equivalent if they can be transformed into each other by a sequence of Kempe changes which interchange the colors used on a component of the subgraph induced by two color classes. Fisk showed that every two vertex 4-colorings of a 3-colorable triangulation on the sphere are Kempe equivalent, and then Mohar extended this result to any 3-colorable planar graph. Fisk also verified that there are 4-chromatic triangulations on the sphere and 3-colorable triangulations on the torus such that some two 4-colorings of them are not Kempe equivalent. In this paper, we show that every two vertex 4-colorings of a 3-colorable triangulation on the projective plane or the Klein bottle are Kempe equivalent. Our result is best possible in terms of all conditions; 3-colorability, the genus of a non-orientable surface, a triangulation (i.e., it cannot be replaced with a graph).

如果一个图的两个顶点着色可以通过Kempe变换序列相互转换，则它们是Kempe等价的，Kempe变换序列交换了由两个颜色类引起的子图组件上使用的颜色。Fisk证明了球面上3色三角剖分的每两个顶点4色都是Kempe等价的，然后Mohar将这一结果推广到任意3色平面图。Fisk还验证了球面上存在4色三角剖分，环面上存在3色三角剖分，使得其中的一些4色剖分不是Kempe等价的。本文证明了在投影平面或克莱因瓶上的可三色三角剖分的每两个顶点四色都是Kempe等价的。我们的结果在所有条件下都是最好的；3色性，不可定向曲面的属，三角剖分（即不能用图代替）。

引用次数: 0

Construction of large cyclic subspace codes via Sidon spaces with dimensions k and k + 1 维数为k和k的Sidon空间构造大循环子空间码 +

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-16 DOI: 10.1016/j.disc.2026.115008

Yongfeng Niu , Chenyu Zhang , Yansheng Wu , Fagang Li

The research on cyclic subspace codes aims to design coding schemes with larger cardinality and optimal minimum distance to meet the demands of modern communication systems for efficient and reliable coding. This paper investigates the construction problem of cyclic subspace codes, utilizing combinatorial numbers to select the exponent of the irreducible element γ to construct different Sidon spaces, including k-dimensional and

(k + 1)

-dimensional spaces. Subsequently, we consider merging the subspace codes generated by these Sidon spaces, resulting in cyclic subspace codes with larger cardinality. Our construction method effectively increases the cardinality of the code while ensuring optimal minimum distance.

循环子空间码的研究旨在设计具有更大基数和最优最小距离的编码方案，以满足现代通信系统对高效可靠编码的要求。本文研究了循环子空间码的构造问题，利用组合数选择不可约元素γ的指数来构造不同的西顿空间，包括k维和（k+1）维空间。随后，我们考虑合并由这些西顿空间生成的子空间码，得到具有更大基数的循环子空间码。我们的构造方法在保证最优最小距离的同时有效地增加了代码的基数。

引用次数: 0

Proof of a conjecture of Green and Liebeck on codes in symmetric groups 关于对称群中码的Green和Liebeck猜想的证明

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-15 DOI: 10.1016/j.disc.2026.114999

Teng Fang, Jinbao Li

<div><div>Let A and B be subsets of a finite group G and r a positive integer. If for every <math><mi>g</mi><mo>∈</mo><mi>G</mi></math>, there are precisely r pairs <math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo><mo>∈</mo><mi>A</mi><mo>×</mo><mi>B</mi></math> such that <math><mi>g</mi><mo>=</mo><mi>a</mi><mi>b</mi></math>, then B is called a code in G with respect to A and we write <math><mi>r</mi><mi>G</mi><mo>=</mo><mi>A</mi><mo>⋅</mo><mi>B</mi></math>. If in addition B is a subgroup of G, then we say that B is a subgroup code in G. In this paper we resolve a conjecture by Green and Liebeck [8, Conjecture 2.3] on certain subgroup codes in the symmetric group <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>. Let <math><mi>n</mi><mo>></mo><mn>2</mn><mi>k</mi></math> and let j be such that <math><msup><mrow><mn>2</mn></mrow><mrow><mi>j</mi></mrow></msup><mo>⩽</mo><mi>k</mi><mo><</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msup></math>. Suppose that X is a conjugacy class in <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math> containing x, and <math><msub><mrow><mi>Y</mi></mrow><mrow><mi>k</mi></mrow></msub></math> is the subgroup <math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>×</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub></math> of <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math>, where the factor <math><msub><mrow><mi>S</mi></mrow><mrow><mi>k</mi></mrow></msub></math> permutes the subset <math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></math> and the factor <math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msub></math> permutes the subset <math><mo>{</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math>. We prove that <math><mi>r</mi><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mi>X</mi><mo>⋅</mo><msub><mrow><mi>Y</mi></mrow><mrow><mi>k</mi></mrow></msub></math> for some positive integer r if and only if the cycle type of x has exactly one cycle of length <math><msup><mrow><mn>2</mn></mrow><mrow><mi>i</mi></mrow></msup></math> for <math><mn>0</mn><mo>⩽</mo><mi>i</mi><mo>⩽</mo><mi>j</mi></math> and all other cycles have length at least <math><mi>k</mi><mo>+</mo><mn>1</mn></math>. We also propose several problems concerning the existence of certain subgroup codes in a finite group G with respect to a conjugation-closed subset i

设A和B是有限群G的子集，r是正整数。如果对于每一个g∈g，恰好有r对(a,b)∈A×B使得g=ab，那么b就被称为g中关于a的一个码，我们写成rG= a·b。如果另外B是G的子群，则我们说B是G中的子群码。本文解决了Green和Liebeck[8，猜想2.3]关于对称群Sn中某些子群码的一个猜想。设n>；2k和j满足2j≤k<；2j+1。设X是Sn中包含X的共轭类，Yk是Sn的子群Sk×Sn−k，其中因子Sk置换子集{1，…，k}，因子Sn−k置换子集{k+1，…，n}。证明对于正整数r， rSn=X⋅Yk当且仅当X的循环类型恰好有一个长度为2i的循环，且对于0≤i≤j，所有其他循环的长度至少为k+1。对于G中的共轭闭子集，给出了有限群G中某些子群码的存在性问题。

引用次数: 0

Matrix product structure of left generalized quaternion group codes 左广义四元数群码的矩阵积结构

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-14 DOI: 10.1016/j.disc.2026.114977

Yuan Cao , Yonglin Cao , Yanyan Gao , Fanghui Ma , Qin Yue

<div><div>Let <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> be the finite field of q elements. For any integer <math><mi>n</mi><mo>≥</mo><mn>2</mn></math>, let <math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math> be the generalized quaternion group of 4n elements and let <math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math> be the dihedral group of 2n elements. Then left ideals of the group algebra <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub><mo>]</mo></math> (resp. <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub><mo>[</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub><mo>]</mo></math>) are called left generalized quaternion group codes (resp. left dihedral codes) over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> of length 4n (resp. 2n) and abbreviated as left <math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math>-codes (resp. left <math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math>-codes) over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>. In this paper, let q be odd and <math><mrow><mi>gcd</mi></mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>q</mi><mo>)</mo><mo>=</mo><mn>1</mn></math>. We prove that any left <math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math>-code over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> is permutation equivalent to a matrix product code by a unique left <math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math>-code and a unique left twisted <math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math>-code over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>. Then we give a precise representation of left twisted <math><msub><mrow><mi>D</mi></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msub></math>-codes over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math> and determine all distinct self-dual codes, self-orthogonal codes and linear complementary dual (LCD) codes among these codes. Hence, we obtain a complete enumeration of all distinct self-orthogonal codes and LCD codes among left <math><msub><mrow><mi>Q</mi></mrow><mrow><mn>4</mn><mi>n</mi></mrow></msub></math>-codes over <math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math>. As applications, we correct some mistakes in Gao and Yue (2021) [27]</spa

设Fq为q个元素的有限域。对于任意整数n≥2，设Q4n为4n个元的广义四元数群，设D2n为2n个元的二面体群。然后群代数Fq[Q4n]的左理想。Fq[D2n])称为左广义四元数群码。左二面体码)除以长度为4n的Fq。2n)，缩写为左Q4n-codes(参见。左D2n-codes) / Fq。本文设q为奇数，且gcd(n,q)=1。通过一个唯一的左d2n码和一个唯一的左扭曲d2n码，证明了Fq上的任何左q4n码都是矩阵积码的置换等价。然后给出了Fq上左旋d2n码的精确表示，并确定了这些码中所有不同的自对偶码、自正交码和线性互补对偶码（LCD）。因此，我们得到了Fq上左q4n码中所有不同的自正交码和LCD码的完整枚举。作为应用，我们纠正了Gao和Yue(2021)[27]中的一些错误，并给出了几个数值例子。

引用次数: 0

Density of 2-critical signed graphs 2临界符号图的密度

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-14 DOI: 10.1016/j.disc.2026.114998

Reza Naserasr , Weiqiang Yu

A balanced k-coloring of a signed graph

(G, σ)

which has no negative loop is to partition its vertices into k sets each of which induces a balanced subgraph, that is a subgraph with no negative cycle. The notion, through basic graph operation, captures the classic proper coloring of graphs as special case.

Having observed the importance of balanced 2-coloring, in this work we study structural conditions which permit a signed graph to admit a 2-coloring. More precisely, slightly modifying the notion of color-critical, we say a signed graph is k-critical if it admits no balanced k-coloring but every proper subgraph of it admits such a coloring.

We show that if

(G, σ)

is a 2-critical signed graph whose underlying graph is not

K_{5}

or an odd cycle, then

| E (G) | \geq \frac{21 | V (G) | - 2 d + 1}{10}

where d is the maximum number of vertex disjoint digons in

(G, σ)

. As a corollary we conclude that, except for the signed graph

(K_{5}, -)

, any signed simple graph with maximum average degree at most 4.2 admits a balanced 2-coloring.

无负环的有符号图（G,σ）的平衡k-着色是将其顶点划分为k个集，每个集引出一个平衡子图，即无负环的子图。这个概念通过基本的图运算，将经典的图的适当着色作为特例。在观察到平衡2-着色的重要性之后，本文研究了允许一个符号图允许2-着色的结构条件。更准确地说，稍微修改一下色临界的概念，我们说一个有符号图是k临界的，如果它不允许平衡的k染色，但它的每个固有子图都允许这样的染色。证明了如果（G,σ）是一个底图不是K5或奇环的2临界符号图，则|E(G)|≥21|V(G)|−2d+110，其中d为（G,σ）中顶点不相交双子的最大个数。作为推论，我们得出，除了有符号图（K5,−）外，任何最大平均度不超过4.2的有符号简单图都是平衡的2-着色。

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引用次数: 0

Antipodality of spherical designs with odd harmonic indices 奇调和指数球形设计的反对性

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-14 DOI: 10.1016/j.disc.2026.114978

Ryutaro Misawa , Akihiro Munemasa , Masanori Sawa

We determine the smallest size of a non-antipodal spherical design with harmonic indices

{1, 3, \dots, 2 m - 1}

to be

2 m + 1

, where m is a positive integer. This is achieved by proving an analogous result for interval designs.

我们确定具有谐波指数{1,3，…，2m−1}的非对对球设计的最小尺寸为2m+1，其中m为正整数。这是通过证明区间设计的类似结果来实现的。

引用次数: 0

s-almost cross-t-intersecting families for finite sets 有限集的s-几乎交叉相交族

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-14 DOI: 10.1016/j.disc.2026.114982

Dehai Liu , Kaishun Wang , Tian Yao

Two families

F

and

G

of k-subsets of an n-set are called s-almost cross-t-intersecting if each member in

F

(resp.

G

) is t-disjoint with at most s members in

G

(resp.

F

). In this paper, we characterize the s-almost cross-t-intersecting families with the maximum product of their sizes. Furthermore, we provide a corresponding stability result after studying the s-almost cross-t-intersecting families which are not cross-t-intersecting.

两个族F和G （n集合的k个子集）被称为s-几乎交叉相交，如果F中的每个元素都对应。G)是t不相交的，在G (p)中最多有s个元素。F)。在本文中，我们用其大小的最大积刻画了s-几乎交叉相交族。此外，我们在研究了不相交的s-几乎相交族后，给出了相应的稳定性结果。

引用次数: 0

Equitable list coloring of sparse graphs 稀疏图的公平表着色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-13 DOI: 10.1016/j.disc.2026.114981

H.A. Kierstead , Alexandr Kostochka , Zimu Xiang

A proper vertex coloring of a graph is equitable if the sizes of all color classes differ by at most 1. For a list assignment L of k colors to each vertex of an n-vertex graph G, an equitable L-coloring of G is a proper coloring of vertices of G from their lists such that no color is used more than

⌈ n / k ⌉

times. Call a graph equitably k-choosable if it has an equitable L-coloring for every k-list assignment L. A graph G is

(a, b)

-sparse if for every

A \subseteq V (G)

, the number of edges in the subgraph

G [A]

of G induced by A is at most

a | A | + b

Our first main result is that every

(\frac{7}{6}, \frac{1}{3})

-sparse graph with minimum degree at least 2 is equitably 3-colorable and equitably 3-choosable. This is sharp. Our second main result is that every

(\frac{5}{4}, \frac{1}{2})

-sparse graph with minimum degree at least 2 is equitably 4-colorable and equitably 4-choosable. This is also sharp.

One of the tools in the proof is the new notion of strongly equitable (SE) list coloring. This notion is both stronger and more natural than equitable list coloring; and our upper bounds are for SE list coloring.

如果所有颜色类的大小相差不超过1，则图的适当顶点着色是公平的。对于为n顶点图G的每个顶点分配L (k)种颜色的列表，G的公平L染色是对G的顶点进行适当的染色，使得任何颜色的使用次数都不超过≤≤n/k²次。如果对每一个k表分配l都有公平的l着色，则称图G是(a,b)-稀疏的，如果对每一个a V(G)，由a引出的G的子图G[a]的边数最多为1个| a| +b。我们的第一个主要结果是，每个最小度至少为2的(76,13)-稀疏图都是可均匀3色和可均匀3选的。这是锋利的。我们的第二个主要结果是，每个最小度至少为2的(54,12)-稀疏图都是可均匀4色和可均匀4选的。这也是尖锐的。证明中的一个工具是强公平表着色的新概念。这个概念比公平的列表着色更强大、更自然；我们的上界是针对SE表着色的。

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引用次数: 0

A constructive characterization of 4-connected 4-regular graphs 4连通4正则图的构造刻划

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-01-13 DOI: 10.1016/j.disc.2026.114997

Kiyoshi Ando , Yoshimi Egawa

In this paper, we give a constructive characterization of 4-connected 4-regular graphs. Two edges of a graph are said to be “independent” if they have no common end vertex. Let G be a 4-connected 4-regular graph. We consider the following three operations on G: choose two independent edges of G, subdivide once, and identify the two new vertices (we call this operation “edge-binding”); delete a vertex x from G, add

K_{4}

G - x

, and add a perfect matching between

V (K_{4})

and

N_{G} (x)

(we call this operation “

K_{4}

-expanding”); delete two independent edges

e_{1}

and

e_{2}

from G, add

K_{4}

G - e_{1} - e_{2}

, and add a perfect matching between

V (K_{4})

and

V (e_{1}) \cup V (e_{2})

(we call this operation “

K_{4}

-edge-binding”). In this paper, we prove that every 4-connected 4-regular graph can be obtained from

K_{5}

K_{4, 4}

by repeated applications of edge-bindings,

K_{4}

-expandings and

K_{4}

-edge-bindings.

本文给出了4连通4正则图的一个构造刻划。如果一个图的两条边没有共同的端点，则称它们是“独立的”。设G是一个4连通的4正则图。我们考虑对G进行以下三种操作：选择G的两条独立边，细分一次，确定两个新的顶点（我们称此操作为“边绑定”）；从G中删除一个顶点x，将K4添加到G−x中，并在V（K4）和NG(x)之间添加一个完美匹配（我们称此操作为“K4扩展”）；从G中删除两条独立边e1和e2，将K4添加到G−e1−e2中，并在V（K4）和V(e1)∪V（e2）之间添加一个完美匹配（我们称此操作为“K4边绑定”）。通过边绑定、K4-展开和K4-边绑定的重复应用，证明了每一个4连通的4正则图都可以由K5或k4,4得到。

引用次数: 0

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