Discrete Mathematics最新文献

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The Mostar index of Tribonacci cubes Tribonacci 立方体的莫斯塔尔指数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-09 DOI: 10.1016/j.disc.2024.114281

Yu Wang, Min Niu

Tribonacci cubes

Γ_{n}^{(3)}

are a class of hypercube-like cubes obtained by removing all vertices of hypercubes

Q_{n}

that have more than two consecutive 1s. In this paper, we calculate the Mostar index of Tribonacci cubes, which is a measure of how far the graph is from being distance-balanced and is used to study various properties of chemical graphs.

Tribonacci 立方体Γn(3) 是一类类似超立方体的立方体，它是通过删除超立方体 Qn 中所有连续出现两个以上 1 的顶点而得到的。本文计算 Tribonacci 立方体的莫斯塔尔指数，该指数是衡量图形距离距离平衡程度的指标，用于研究化学图形的各种性质。

引用次数: 0

Structural parameters of Schnyder woods 施奈德木材的结构参数

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-09 DOI: 10.1016/j.disc.2024.114282

Christian Ortlieb , Jens M. Schmidt

We study two fundamental parameters of Schnyder woods by exploiting structurally related methods. First, we prove a new lower bound on the total number of leaves in the three trees of a Schnyder wood. Second, it is well-known that Schnyder woods can be used to find three compatible ordered path partitions. We prove new lower bounds on the number of singletons, i.e. paths that consists of exactly one vertex, in such compatible ordered path partitions. All bounds that we present are tight.

我们利用结构相关的方法研究了施奈德林的两个基本参数。首先，我们证明了施奈德林三棵树中叶子总数的新下限。其次，众所周知，施奈德林可以用来寻找三个兼容的有序路径分区。我们证明了这种兼容有序路径分区中单子（即只包含一个顶点的路径）数量的新下界。我们提出的所有界限都很严密。

引用次数: 0

On isometry and equivalence of skew constacyclic codes 论倾斜常环码的等同性和等价性

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-09 DOI: 10.1016/j.disc.2024.114279

Hassan Ou-azzou , Mustapha Najmeddine , Nuh Aydin

In this paper we generalize the notion of n-isometry and n-equivalence relation introduced by Chen et al. in [13], [12] to classify constacyclic codes of length n over a finite field

F_{_{q}}

, where

q = p^{r}

is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively

(n, σ)

-equivalence and

(n, σ)

-isometric relation, where n is the length of the code and σ is an automorphism of the finite field. We compute the number of

(n, σ)

-equivalence and

(n, σ)

-isometric classes, and we give conditions on λ and μ for which

(σ, λ)

-constacyclic codes and

(σ, μ)

-constacyclic codes are equivalent. Under some conditions on n and q we prove that skew constacyclic codes are equivalent to cyclic codes by using properties of our equivalence relation introduced. We also prove that when q is even and σ is the Frobenius automorphism, skew constacyclic codes of length n are equivalent to cyclic codes when

\gcd (n, r) = 1

. Finally we give some examples as applications of the theory developed here.

在本文中，我们将 Chen 等人在 [13], [12] 中引入的 n 等式关系和 n 等价关系的概念，用于对有限域 Fq（其中 q=pr 是素幂次）上长度为 n 的共环码进行分类，并将其推广到无派生的倾斜共环码的情况中。我们把这些关系分别称为（n,σ）-等价关系和（n,σ）-等距关系，其中 n 是码的长度，σ 是有限域的自变量。我们计算了(n,σ)等价类和(n,σ)等距类的数量，并给出了λ和μ的条件，在这些条件下，(σ,λ)等价编码和(σ,μ)等价编码是等价的。在 n 和 q 的某些条件下，我们利用所引入的等价关系的性质，证明了倾斜常环码等价于循环码。我们还证明，当 q 为偶数且 σ 为弗罗贝尼斯自变分时，长度为 n 的偏斜自循环码等价于 gcd(n,r)=1 时的循环码。最后，我们将举例说明本文理论的应用。

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引用次数: 0

On cyclic symmetric Hamilton cycle decompositions of complete multipartite graphs 论完整多方图的循环对称汉密尔顿循环分解

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-08 DOI: 10.1016/j.disc.2024.114277

Fatima Akinola , Michael W. Schroeder

A decomposition of a graph with n vertices, labeled by

Z_{n}

, is cyclic if addition by 1 to the vertices acts on the decomposition, and the decomposition is d-symmetric for a divisor d of n if addition by

n / d

to the vertices acts invariantly on the decomposition. In a 2017 paper, Merola et al. established the necessary and sufficient conditions under which a complete multipartite graph with an even number of parts, each with d vertices, has a cyclic Hamilton cycle decomposition; these decompositions were also d-symmetric.

In this paper we establish the necessary and sufficient conditions for the analogous question with complete multipartite graphs with an odd number of parts, which settles the existence of cyclic, d-symmetric Hamilton cycle decompositions for all balanced, complete multipartite graphs.

一个有 n 个顶点的图的分解（用 Zn 标记），如果顶点的加法 1 作用于该分解，则该分解是循环的；如果顶点的加法 n/d 不变地作用于该分解，则该分解对于 n 的除数 d 是 d 对称的。在 2017 年的一篇论文中，Merola 等人确定了偶数部分的完整多部分图（每个部分有 d 个顶点）具有循环汉密尔顿循环分解的必要条件和充分条件；这些分解也是 d 对称的。在本文中，我们为奇数部分的完整多部分图的类似问题确定了必要条件和充分条件，从而解决了所有平衡完整多部分图的循环、d 对称汉密尔顿循环分解的存在性问题。

引用次数: 0

On the proper interval completion problem within some chordal subclasses 关于某些和弦子类中的适当音程完成问题

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-07 DOI: 10.1016/j.disc.2024.114274

François Dross , Claire Hilaire , Ivo Koch , Valeria Leoni , Nina Pardal , María Inés Lopez Pujato , Vinicius Fernandes dos Santos

Given a property (graph class) Π, a graph G, and an integer k, the Π-completion problem consists of deciding whether we can turn G into a graph with the property Π by adding at most k edges to G. The Π-completion problem is known to be NP-hard for general graphs when Π is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.

给定一个属性（图类）Π、一个图 G 和一个整数 k，Π-补全问题包括判断我们是否能通过在 G 上添加最多 k 条边将 G 变成一个具有属性 Π 的图。众所周知，当 Π 是适当区间图 (PIG) 的属性时，Π-补全问题对于一般图来说是 NP-困难的。在这项工作中，我们研究了弦图不同子类中的 PIG-补全问题。我们证明，即使仅限于分裂图，该问题仍然是 NP-完全的。然后，我们将注意力转向正面结果，并提出了多项式时间算法，用于解决输入仅限于毛毛虫图和阈值图时的 PIG-补全问题。我们还提出了准阈值图的最小共边完成的高效算法，为该图类中的 PIG 完成问题提供了一个下界。

引用次数: 0

Octopuses in the Boolean cube: Families with pairwise small intersections, part II 布尔立方中的章鱼：具有成对小交集的族，第二部分

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-07 DOI: 10.1016/j.disc.2024.114280

Andrey Kupavskii , Fedor Noskov

The problem we consider originally arises from 2-level polytope theory. This class of polytopes generalizes a number of other polytope families. One of the important questions in this field can be formulated as follows: is it true for a d-dimensional 2-level polytope that the product of the number of its vertices and the number of its

d - 1

dimensional facets is bounded by

d 2^{d - 1}

? Recently, Kupavskii and Weltge [9] settled this question in positive. A key element in their proof is a more general result for families of vectors in

R^{d}

such that the scalar product between any two vectors from different families is either 0 or 1.

Peter Frankl noted that, when restricted to the Boolean cube, the solution boils down to an elegant application of the Harris–Kleitman correlation inequality. Meanwhile, this problem becomes much more sophisticated when we consider several families.

Let

F_{1}, \dots, F_{ℓ}

be families of subsets of

{1, \dots, n}

. We suppose that for distinct

k, k^{'}

and arbitrary

F_{1} \in F_{k}, F_{2} \in F_{k^{'}}

we have

| F_{1} \cap F_{2} | ⩽ m

. We are interested in the maximal value of

| F_{1} | \dots | F_{ℓ} |

and the structure of the extremal example.

In the previous paper on the topic, the authors found the asymptotics of this product for constant ℓ and m as n tends to infinity. However, the possible structure of the families from the extremal example turned out to be very complicated. In this paper, we obtain a strong structural result for the extremal families.

我们所考虑的问题最初源于 2 级多面体理论。这一类多面体概括了许多其他多面体族。该领域的一个重要问题可以表述如下：对于一个 d 维 2 层多面体，其顶点数与 d-1 维面数的乘积是否真的以 d2d-1 为界？最近，Kupavskii 和 Weltge [9] 从正面解决了这个问题。彼得-弗兰克尔（Peter Frankl）指出，当局限于布尔立方体时，解决方法可以归结为哈里斯-克莱特曼相关不等式的优雅应用。让 F1、......、Fℓ 分别是 {1，......，n} 的子集族。我们假设，对于不同的 k,k′ 和任意的 F1∈Fk,F2∈Fk′，我们有 |F1∩F2|⩽m。我们感兴趣的是|F1|...|Fℓ|的最大值和极值实例的结构。在上一篇相关论文中，作者发现了当 n 趋于无穷大时，常数 ℓ 和 m 的该积的渐近线。然而，从极值示例中得出的族的可能结构却非常复杂。在本文中，我们得到了极值族的强结构结果。

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引用次数: 0

Turán numbers and switching 图兰数字和转换

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-07 DOI: 10.1016/j.disc.2024.114275

Karen Gunderson , Jason Semeraro

Using a switching operation on tournaments we obtain some new lower bounds on the Turán number of the r-graph on

r + 1

vertices with 3 edges. For

r = 4

, extremal examples were constructed using Paley tournaments in previous work. We show that these examples are unique (in a particular sense) using Fourier analysis.

A 3-tournament is a ‘higher order’ version of a tournament given by an alternating function on triples of distinct vertices in a vertex set. We show that 3-tournaments also enjoy a switching operation and use this to give a formula for the size of a switching class in terms of level permutations, generalising a result of Babai–Cameron.

利用锦标赛的切换操作，我们获得了 r+1 个顶点上有 3 条边的 r 图的图兰数的一些新下限。对于 r=4，我们在之前的工作中利用帕利锦标赛构建了极值实例。我们利用傅立叶分析法证明这些例子是唯一的（在特定意义上）。3-锦标赛是锦标赛的 "高阶 "版本，由顶点集合中不同顶点的三元组上的交替函数给出。我们证明了 3 锦标也有切换操作，并利用这一点给出了以水平排列为单位的切换类大小公式，从而推广了巴拜-卡梅隆的一个结果。

引用次数: 0

The edge coloring of the Cartesian product of signed graphs 有符号图形笛卡尔积的边着色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-04 DOI: 10.1016/j.disc.2024.114276

Chao Wen , Qiang Sun , Hongyan Cai , Chao Zhang

According to Vizing's Theorem, a major question in the area of edge coloring is to determine whether a graph is Class 1 or 2. In 1984, Mohar proved that the Cartesian product

G □ H

is Class 1 if G is Class 1 or both G and H have a perfect matching. Recently, Behr proved that the signed graph version of Vizing's Theorem: a signed graph

(G, σ)

is either Class 1 or 2. Hence, we want to generalize Mohar's results to signed graphs. In this paper, we prove that

(G, σ) □ (H, π)

is Class 1 if one of the factors, say

(G, σ)

, is Class 1 and there exists an edge coloring of

(G, σ)

that satisfies a certain property, which is necessary as shown by an example. Let Δ-matching be a matching which covers every vertex of maximum degree. We also show that if both of

(G, σ)

and

(H, π)

have a Δ-matching and at least one of

Δ (G), Δ (H)

is even, then

(G, σ) □ (H, π)

is Class 1. This implies that if both of G and H have a Δ-matching, then

G □ H

is Class 1, thereby slightly improving upon Mohar's results.

根据维京定理，边着色领域的一个主要问题是确定一个图是第 1 类还是第 2 类。1984 年，莫哈尔证明，如果 G 是第 1 类图，或者 G 和 H 都有完美匹配，则笛卡尔积 G□H 是第 1 类图。最近，Behr 证明了 Vizing 定理的有符号图版本：有符号图 (G,σ) 要么是第 1 类，要么是第 2 类。因此，我们希望将莫哈尔的结果推广到有符号图。在本文中，我们将证明，如果其中一个因子（比如说 (G,σ)）是第 1 类，并且 (G,σ) 的边着色满足某个属性，则 (G,σ)□(H,π) 是第 1 类。假设 Δ-matching 是一个覆盖了最大度顶点的匹配。我们还证明，如果（G,σ）和（H,π）都有Δ匹配，并且Δ(G),Δ(H)中至少有一个是偶数，那么（G,σ）□（H,π）就是第 1 类。这意味着如果 G 和 H 都有Δ匹配，那么 G□H 是第 1 类，从而稍微改进了莫哈尔的结果。

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引用次数: 0

Linear colouring of binomial random graphs 二项式随机图形的线性着色

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-03 DOI: 10.1016/j.disc.2024.114278

Austin Eide, Paweł Prałat

We investigate the linear chromatic number

χ_{lin} (G (n, p))

of the binomial random graph

G (n, p)

on n vertices in which each edge appears independently with probability

p = p (n)

. For a graph G,

χ_{lin} (G)

is defined as the smallest k such that G admits a k-colouring with the property that every path P in G receives a colour which appears on only one vertex of P. For dense random graphs (

n p \to \infty

n \to \infty

), we show that asymptotically almost surely

χ_{lin} (G (n, p)) \geq n (1 - O ({(n p)}^{- 1 / 2})) = n (1 - o (1))

. Understanding the order of the linear chromatic number for subcritical random graphs (

n p < 1

) and critical ones (

n p = 1

) is relatively easy. However, supercritical sparse random graphs (

n p = c

for some constant

c > 1

) remain to be investigated.

我们研究了 n 个顶点上的二项式随机图 G(n,p)的线性色度数 χlin(G(n,p))，其中每条边都以 p=p(n) 的概率独立出现。对于一个图 G，χlin(G) 被定义为最小的 k，使得 G 可以接受 k-着色，其特性是 G 中的每条路径 P 得到的颜色只出现在 P 的一个顶点上。对于密集随机图（np→∞ 为 n→∞），我们证明了渐近几乎肯定 χlin(G(n,p))≥n(1-O((np)-1/2))=n(1-o(1))。理解亚临界随机图（np<1）和临界随机图（np=1）的线性色度数阶相对容易。然而，超临界稀疏随机图（np=c，对于某个常数 c>1）仍有待研究。

引用次数: 0

The Collatz map analogue in polynomial rings and in completions 多项式环和完备中的科拉茨图类似物

IF 0.7 3区数学 Q2 MATHEMATICS

Discrete Mathematics

Pub Date : 2024-10-03 DOI: 10.1016/j.disc.2024.114273

Angelot Behajaina , Elad Paran

We study an analogue of the Collatz map in the polynomial ring

R [x]

, where R is an arbitrary commutative ring. We prove that if R is of positive characteristic, then every polynomial in

R [x]

is eventually periodic with respect to this map. This extends previous works of the authors and of Hicks, Mullen, Yucas and Zavislak, who studied the Collatz map on

F_{p} [x]

and

F_{2} [x]

, respectively. We also consider the Collatz map on the ring of formal power series

R [[x]]

when R is finite: we characterize the eventually periodic series in this ring, and give formulas for the number of cycles induced by the Collatz map, of any given length. We provide similar formulas for the original Collatz map defined on the ring

Z_{2}

of 2-adic integers, extending previous results of Lagarias.

我们研究了多项式环 R[x] 中的科拉茨映射，其中 R 是任意交换环。我们证明，如果 R 是正特征，那么 R[x] 中的每个多项式最终都是关于这个映射的周期性多项式。这扩展了作者以及希克斯、马伦、尤卡斯和扎维斯拉克之前的工作，他们分别研究了 Fp[x] 和 F2[x] 上的科拉茨映射。我们还考虑了当 R 有限时形式幂级数环 R[[x]] 上的科拉茨映射：我们描述了该环中最终周期数列的特征，并给出了任何给定长度的科拉茨映射诱导的循环数公式。我们为定义在二阶整数环 Z2 上的原始科拉茨映射提供了类似的公式，扩展了拉加里亚斯以前的结果。

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引用次数: 0

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