Discrete Mathematics最新文献

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Maximum size of connected graphs with bounded maximum degree and matching number 具有有界最大度和匹配数的连通图的最大尺寸

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-07-01 Epub Date: 2026-01-29 DOI: 10.1016/j.disc.2026.115019

Zixuan Yang , Hongliang Lu , Shenggui Zhang

In this paper, we determine the maximum number of edges of connected graphs with given maximum degree and matching number. This gives an answer to a problem posed by Dibek et al. (2017) [6]. We also show that the bound in our result is tight.

在给定最大度和匹配数的情况下，确定连通图的最大边数。这就回答了Dibek等人（2017）提出的问题。我们还证明了结果中的界是紧的。

引用次数: 0

Boxicity and cubicity of a subclass of divisor graphs and power graphs of cyclic groups 循环群的除数图和幂图一类的有利性和立方性

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-07-01 Epub Date: 2026-01-28 DOI: 10.1016/j.disc.2026.115017

L. Sunil Chandran , Jinia Ghosh

The boxicity (respectively, cubicity) of an undirected graph Γ is the smallest non-negative integer k such that Γ can be represented as the intersection graph of axis-parallel rectangular boxes (respectively, unit cubes) in

R^{k}

. An undirected graph is classified as a comparability graph if it is isomorphic to the comparability graph of some partial order. In this paper, we initiate the study of boxicity and cubicity for two subclasses of comparability graphs - divisor graphs and power graphs.

Divisor graphs, an important family of comparability graphs, arise from a number-theoretically defined poset, namely the divisibility poset. We consider one of the most popular subclasses of divisor graphs, denoted by

D (n)

, where the vertex set is the set of positive divisors of a natural number n, and two vertices a and b are adjacent if and only if a divides b or b divides a. We derive estimates, tight up to a factor of 2, for the boxicity and cubicity of

D (n)

Power graphs are a special class of algebraically defined comparability graphs. The power graph of a group is an undirected graph whose vertex set is the group itself, with two elements being adjacent if one is a power of the other. We show that studying the boxicity (respectively, cubicity) of

D (n)

is sufficient to study the boxicity (respectively, cubicity) of the power graph of the cyclic group of order n. Thus, as a corollary of our first result, we derive similar estimates for the boxicity and cubicity power graphs of cyclic groups.

无向图Γ的有利度（即立方度）是最小的非负整数k，使得Γ可以表示为Rk中坐标轴平行的矩形框（即单位立方体）的相交图。如果无向图与某偏阶的可比性图同构，则将其分类为可比性图。本文研究了可比性图的两个子类——除数图和幂图的有利性和立方性。除数图是一类重要的可比性图，它产生于一个数论定义的偏序集，即可除偏序集。我们考虑最流行的一个子类的除数图，表示为D(n)，其中顶点集是自然数n的正除数集，两个顶点a和b相邻当且仅当a除b或b除a。我们导出估计，紧到2的因子，为D(n)的有害性和立方性。幂图是一类特殊的代数定义的可比性图。群的幂图是一个无向图，其顶点集是群本身，如果一个元素是另一个元素的幂，则两个元素相邻。我们证明，研究D(n)的毒性（分别，立方度）足以研究n阶循环群的幂图的毒性（分别，立方度）。因此，作为第一个结果的推论，我们得到了循环群的毒性和立方度幂图的类似估计。

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引用次数: 0

The neighbor graph of self-dual codes over the ring of integers modulo 4 以4为模的整数环上的自对偶码的邻居图

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-07-01 Epub Date: 2026-01-28 DOI: 10.1016/j.disc.2026.115018

Steven T. Dougherty , Esengül Saltürk

We describe the neighbor construction for self-dual codes over

Z_{4}

and give the type of the neighbor based on the type of the code and vector v used to construct the neighbor. We define the neighbor graph of self-dual codes over

Z_{4}

as the graph whose vertices are the self-dual codes of length n and two codes are connected if one can be constructed from the other by the neighbor construction. We show that this graph is connected and regular with degree

2 (\sum_{k = 1}^{⌊ \frac{n}{4} ⌋} C (n, 4 k))

我们描述了Z4上自对偶码的邻域构造，并根据编码的类型和构造邻域的向量v给出了邻域的类型。我们将Z4上的自对偶码的邻居图定义为顶点为长度为n的自对偶码，且两个码之间可以通过邻居构造相互连通的图。我们证明了这个图是连通的、正则的，其次数为2（∑k=1⌊n4⌋C(n,4k)）。

引用次数: 0

A further investigation on covering systems with odd moduli 奇模覆盖系统的进一步研究

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-07-01 Epub Date: 2026-01-28 DOI: 10.1016/j.disc.2026.115013

Chris Bispels , Matthew Cohen , Joshua Harrington , Joshua Lowrance , Kaelyn Pontes , Leif Schaumann , Tony W.H. Wong

Erdős first introduced the idea of covering systems in 1950. Since then, much of the work in this area has concentrated on identifying covering systems that meet specific conditions on their moduli. Among the central open problems in this field is the well-known odd covering problem. In this paper, we investigate a variant of that problem, where one odd integer is permitted to appear multiple times as a modulus in the covering system, while all remaining moduli are distinct odd integers greater than 1.

Erdős在1950年首次提出了覆盖系统的概念。从那时起，该领域的大部分工作都集中在识别满足其模的特定条件的覆盖系统上。该领域的核心开放问题之一是著名的奇覆盖问题。本文研究了该问题的一个变体，允许一个奇数作为模在覆盖系统中出现多次，而其余的模都是大于1的不同奇数。

引用次数: 0

Spectral Turán-type problems on sparse spanning graphs 稀疏生成图上的谱Turán-type问题

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-07-01 Epub Date: 2026-01-27 DOI: 10.1016/j.disc.2026.115016

Lele Liu , Bo Ning

<div><div>Let F be a graph, and let <math><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math>, <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math>, and <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> denote the classes of graphs that attain, respectively, the maximum number of edges, the maximum adjacency spectral radius, and the maximum signless Laplacian spectral radius over all n-vertex graphs that do not contain F as a subgraph. A fundamental problem in spectral extremal graph theory is to characterize all graphs F for which <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> when n is sufficiently large. Establishing the conjecture of Cioabă et al. (2022) [10], Wang et al. (2023) [54] proved that: for any graph F such that the graphs in <math><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> are Turán graphs plus <math><mi>O</mi><mo>(</mo><mn>1</mn><mo>)</mo></math> edges, <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> for sufficiently large n. In addition, another interesting problem in spectral extremal graph theory is to characterize all graphs F such that <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> for sufficiently large n.</div><div>In this paper, we give new contribution to the problems mentioned above. First, we present a substantial collection of examples of graphs F for which <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>⊆</mo><mi>EX</mi><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></math> holds when n is sufficiently large, focusing on n-vertex graph F with no isolated vertices and maximum degree <math><mi>Δ</mi><mo>(</mo><mi>F</mi><mo>)</mo><mo>≤</mo><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>/</mo><mn>40</mn></math>. Second, under the same conditions on F, we prove that <math><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>A</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo><mo>=</mo><msub><mrow><mi>SPEX</mi></mrow><mrow><mi>Q</mi></mrow></msub><mo>(</mo><mi

设F是一个图，设EX（n，F）、SPEXA（n，F）和SPEXQ（n，F）分别表示在所有不包含F作为子图的n顶点图上获得最大边数、最大邻接谱半径和最大无符号拉普拉斯谱半径的图的类别。谱极值图论中的一个基本问题是，当n足够大时，对所有的图F （SPEXA(n，F)）进行描述。通过建立cioabu et al. (2022) b[10]的猜想，Wang et al. (2023) b[54]证明：对于任意图F，使得EX（n，F）中的图为Turán图加O(1)条边，对于足够大的n， SPEXA（n，F）的任任任任，对于谱极值图论中的另一个有趣问题是，对所有图F进行描述，使得对于足够大的n， SPEXA(n，F)=SPEXQ（n，F）。本文对上述问题给出了新的贡献。首先，我们给出了大量在n足够大时，SPEXA(n，F)≥≥EX（n，F）成立的图F的例子，重点关注无孤立顶点且最大度数Δ(F)≤n/40的n顶点图F。其次，在F上相同的条件下，我们证明了对于足够大的n， SPEXA(n，F)=SPEXQ（n，F）。这些结果可以看作是Alon和Yuster(2013)[1]定理的谱类比。进一步，作为直接推论，我们得到了几类特殊图存在的紧谱条件，包括团因子、汉密尔顿环的k次幂和图中的k因子。第一类特殊的图给出了Feng的一个问题的正答案，第二类特殊的图扩展了Yan等人之前的结果。

引用次数: 0

Counting degree-constrained orientations 计数受程度约束的方向

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-29 DOI: 10.1016/j.disc.2026.115024

Jing Yu , Jie-Xiang Zhu

We study the enumeration of graph orientations under local degree constraints. Given a finite graph

G = (V, E)

and a family of admissible sets

{P_{v} \subseteq Z : v \in V}

, let

N (G; \prod_{v \in V} P_{v})

denote the number of orientations in which the out-degree of each vertex v lies in

P_{v}

. We prove a general duality formula expressing

N (G; \prod_{v \in V} P_{v})

as a signed sum over edge subsets, involving products of coefficient sums associated with

{P_{v}}_{v \in V}

, from a family of polynomials. Our approach employs gauge transformations, a technique rooted in statistical physics and holographic algorithms. We also present a probabilistic derivation of the same identity, interpreting the orientation-generating polynomial as the expectation of a random polynomial product. As applications, we obtain explicit formulas for the number of even orientations and for mixed Eulerian-even orientations on general graphs. Our formula generalizes a result of Borbényi and Csikvári on Eulerian orientations of graphs.

研究了局部度约束下图的定向枚举问题。给定一个有限图G=（V，E）和一组可容许集合{Pv∈Z: V∈V}，令N（G;∏V∈VPv）表示每个顶点V在Pv中的出度所在的方向个数。我们证明了一个一般对偶公式，将N（G;∏v∈VPv）表示为边缘子集上的有符号和，涉及多项式族中与{Pv}v∈v相关的系数和的乘积。我们的方法采用标准变换，一种植根于统计物理学和全息算法的技术。我们还提出了相同恒等式的概率推导，将方向生成多项式解释为随机多项式乘积的期望。作为应用，我们得到了一般图上偶向数和混合欧拉偶向数的显式公式。我们的公式推广了borb和Csikvári关于图的欧拉取向的结果。

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引用次数: 0

A Metzler matrix of a group covering of a digraph 有向图的群覆盖的Metzler矩阵

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-19 DOI: 10.1016/j.disc.2026.115006

Yusuke Ide , Takashi Komatsu , Norio Konno , Iwao Sato

We present a decomposition formula for the determinant of a Metzler matrix

A (H)

of a group covering H of a digraph D. Furthermore, we introduce an L-function of D with respect to its Metzler matrix

A (D)

, and present a determinant expression of it. As a corollary, we present a decomposition formula for the determinant of a Metzler matrix

A (H)

of a group covering H of D by its L-functions.

给出了有向图D中覆盖H的群的Metzler矩阵a (H)的行列式的分解公式，进一步引入了D关于其Metzler矩阵a (D)的l函数，并给出了它的行列式表达式。作为推论，我们给出了由l函数覆盖H (D)的群的Metzler矩阵a (H)的行列式的分解公式。

引用次数: 0

On DP-coloring of outerplanar graphs 关于外平面图的dp染色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-16 DOI: 10.1016/j.disc.2026.114996

Tianjiao Dai , Jie Hu , Hao Li , Shun-ichi Maezawa

The notion of DP-coloring was introduced by Dvořák and Postle which is a generalization of list coloring. A DP-coloring of a graph G reduces the problem of finding a proper coloring of G from a given list L to the problem of finding a “large” independent set in an auxiliary graph

M_{L}

-cover with a vertex set

{(v, c) : v \in V (G)

and

c \in L (v)}

. Hutchinson (Journal of Graph Theory, 2008) showed that

•
if a 2-connected bipartite outerplanar graph G has a list of colors $L (v)$ for each vertex v with $| L (v) | \geq \min {\deg_{G} (v), 4}$ , then G is L-colorable; and
•
if a 2-connected maximal outerplanar graph G with at least four vertices has a list of colors $L (v)$ for each vertex v with $| L (v) | \geq \min {\deg_{G} (v), 5}$ , then G is L-colorable.

In this paper, we study whether bounds of Hutchinson's results hold for DP-coloring. We obtain that the first one is not sufficient for DP-coloring while the second one is sufficient.

dp -着色的概念是由Dvořák和Postle提出的，它是列表着色的推广。图G的dp -着色将从给定列表L中寻找G的适当着色问题简化为在具有顶点集{(v,c):v∈v (G) and c∈L(v)}的辅助图ML-cover中寻找“大”独立集的问题。Hutchinson （Journal of Graph Theory, 2008）证明了•如果一个2连通二部外平面图G对于每个顶点v有一个颜色列表L(v)且|L(v)|≥min (degG) {degG (v),4}，则G是L可色的；•如果一个至少有四个顶点的2连通最大外平面图G对每个顶点v都有一个颜色列表L(v)，且|L(v)|≥min (degG) (v),5}，则G是L可色的。在本文中，我们研究了Hutchinson结果的界对于dp -着色是否成立。我们得到第一个是不充分的，而第二个是充分的。

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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 : 2026-06-01 Epub 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

A remark on a result on odd colorings of planar graphs 关于平面图奇色的一个结果的注解

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2026-06-01 Epub Date: 2026-01-27 DOI: 10.1016/j.disc.2026.115014

Dinabandhu Pradhan , Vaishali Sharma , Riste Škrekovski

A proper k-coloring of a graph is said to be odd if every non-isolated vertex has a color that appears an odd number of times on its neighborhood. Miao et al. (2024) [2] claimed that every planar graph without adjacent 3-cycles is odd 7-colorable and every triangle-free planar graph without intersecting 4-cycles is odd 5-colorable. Here, we point out that their published proof contains a fundamental flaw which affects the validity of the main results.

如果每个非孤立顶点的颜色在其邻域上出现奇数次，则称图的适当k-着色为奇数。Miao et al.(2024)[2]提出无相邻3环的平面图都是奇7色，无4环相交的无三角形平面图都是奇5色。在此，我们指出他们发表的证明存在一个影响主要结果有效性的根本性缺陷。

引用次数: 0

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