Discrete Applied Mathematics最新文献

英文中文

On nonrepetitive colorings of paths and cycles 论路径和循环的非重复着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-18 DOI: 10.1016/j.dam.2024.08.018

<div>We say that a sequence <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn><mi>t</mi></mrow></msub></mrow></math> of integers is repetitive if <math><mrow><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mi>t</mi></mrow></msub></mrow></math> for every <math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>t</mi><mo>}</mo></mrow></mrow></math>. A walk in a graph <math><mi>G</mi></math> is a sequence <math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>r</mi></mrow></msub></mrow></math> of vertices of <math><mi>G</mi></math> in which <math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math> for every <math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>r</mi><mo>−</mo><mn>1</mn><mo>}</mo></mrow></mrow></math>. Given a <math><mi>k</mi></math>-coloring <math><mrow><mi>c</mi><mo>:</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>→</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math> of <math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, we say that <math><mi>c</mi></math> is walk-nonrepetitive (resp. stroll-nonrepetitive) if for every <math><mrow><mi>t</mi><mo>∈</mo><mi>N</mi></mrow></math> and every walk <math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mi>t</mi></mrow></msub></mrow></math> the sequence <math><mrow><mi>c</mi><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>)</mo></mrow><mo>⋯</mo><mi>c</mi><mrow><mo>(</mo><msub><mrow><mi>v</mi></mrow><mrow><mn>2</mn><mi>t</mi></mrow></msub><mo>)</mo></mrow></mrow></math> is not repetitive unless <math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mo>+</mo><mi>t</mi></mrow></msub></mrow></math> for every <math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>t</mi><mo>}</mo></mrow></mrow></math> (resp. unless <math><mrow><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>=</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi><mo>+</mo><mi>t</mi></mrow></msub></mrow></math> for some <math><mrow><mi>i</mi><mo>∈</mo><mrow><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo>

如果每 i∈{1,...,t}，ai=ai+t，我们就说整数序列 a1⋯a2t 是重复的。图 G 中的行走是 G 的顶点序列 v1⋯vr，其中每 i∈{1，...，r-1}，vivi+1∈E(G)。给定 V(G) 的 k 个着色 c:V(G)→{1,...,k} ，如果 c 是漫步非重复的（resp.对于每个 t∈N 和每个行走 v1⋯v2t 序列 c(v1)⋯c(v2t)都不重复，除非对于每个 i∈{1，...，t}，vi=vi+t（或者，除非对于某些 i∈{1，...，t}，vi=vi+t）。G 的漫步（或漫步）色度数 σ(G)（或 ρ(G)）是 G 具有漫步非重复（或漫步非重复）k 着色的最小 k。让 Cn 和 Pn 分别表示有 n 个顶点的循环和路径。本文提出了三个结果，回答了 Barát 和 Wood 在 2008 年提出的问题：(i) 只要 n≥4 且 n∉{5,7}，σ(Cn)=4；(ii) 如果 3≤n≤21 ρ(Pn)=3，否则 ρ(Pn)=4；(iii) 只要 n∉{3,4,6,8}，ρ(Cn)=4，否则 ρ(Cn)=3。特别是，(ii) 改进了 Ochem 和 Tao 分别于 2021 年和 2023 年得到的 n 定界。

引用次数: 0

Study on geometric–arithmetic, arithmetic–geometric and Randić indices of graphs 图形的几何指数、算术指数和兰迪克指数研究

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-18 DOI: 10.1016/j.dam.2024.09.007

Topological indices are mathematical descriptors used in the field of chemistry to characterize the topological structure of chemical compounds. The Randić index ( $R$ ), the geometric–arithmetic index ( $G A$ ), and the arithmetic–geometric index ( $A G$ ) represent three widely recognized topological indices. In most scenarios, the properties of $A G$ and $G A$ exhibit opposing tendencies. Furthermore, it is observed that, $A G (G) > R (G)$ and $G A (G) > R (G)$ for any given graph $G$ . Our focus is thus directed towards investigating the gaps between $A G$ and $R$ , as well as $G A$ and $R$ . We find that the invariants $A G - R$ and $G A - R$ correlate well with some molecular properties. Numerous upper and lower bounds for the quantities $A G - R$ and $G A - R$ are computed for general graphs, bipartite graphs, chemical graphs, trees, and chemical trees, in terms of graph order, with an emphasis on characterizing extremal graphs.

拓扑指数是化学领域用来描述化合物拓扑结构的数学描述符。兰迪克指数（R）、几何-算术指数（GA）和算术-几何指数（AG）是三种广为认可的拓扑指数。在大多数情况下，AG 和 GA 的属性表现出相反的趋势。此外，我们还观察到，对于任何给定的图 G，AG(G)>R(G)和 GA(G)>R(G)。因此，我们的重点是研究 AG 和 R 以及 GA 和 R 之间的差距。我们根据图的阶数计算了一般图、二叉图、化学图、树和化学树的 AG-R 和 GA-R 量的大量上界和下界，重点是极值图的特征。

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引用次数: 0

On the Aα-index of graphs with given order and dissociation number 关于给定阶数和解离数的图形的 Aα 指数

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-13 DOI: 10.1016/j.dam.2024.09.002

Given a graph $G,$ a subset of vertices is called a maximum dissociation set of $G$ if it induces a subgraph with vertex degree at most 1, and the subset has maximum cardinality. The cardinality of a maximum dissociation set is called the dissociation number of $G$ . The adjacency matrix and the degree diagonal matrix of $G$ are denoted by $A (G)$ and $D (G),$ respectively. In 2017, Nikiforov proposed the $A_{α}$ -matrix: $A_{α} (G) = α D (G) + (1 - α) A (G),$ where $α \in [0, 1] .$ The largest eigenvalue of this novel matrix is called the $A_{α}$ -index of $G .$ In this paper, we firstly determine the connected graph (resp. bipartite graph, tree) having the largest $A_{α}$ -index over all connected graphs (resp. bipartite graphs, trees) with fixed order and dissociation number. Secondly, we describe the structure of all the $n$ -vertex graphs having the minimum $A_{α}$ -index with dissociation number $τ$ , where $τ ⩾ ⌈ \frac{2}{3} n ⌉ .$ Finally, we identify all the connected $n$ -vertex graphs with dissociation number $τ \in {2, ⌈ \frac{2}{3} n ⌉, n - 1, n - 2}$ having the minimum $A_{α}$ -index.

给定一个图 G，如果顶点子集能诱导出顶点度最多为 1 的子图，且该子集具有最大心数，则该顶点子集称为 G 的最大解离集。G 的邻接矩阵和度对角矩阵分别用 A(G) 和 D(G) 表示。2017 年，尼基福罗夫提出了 Aα 矩阵：Aα（G）=αD（G）+（1-α）A（G），其中α∈[0,1]。在本文中，我们首先确定了在所有具有固定阶数和解离数的连通图（即二元图、树）中具有最大 Aα-index 的连通图（即二元图、树）。其次，我们描述了具有最小 Aα 指数且解离数为 τ 的所有 n 顶点图的结构，其中 τ⩾⌈23n⌉.最后，我们确定了所有具有解离数 τ∈{2,⌈23n⌉,n-1,n-2} 的 n 个连接顶点图，这些图具有最小 Aα 指数。

引用次数: 0

A spectral condition for a graph to have strong parity factors 图形具有强奇偶因子的谱条件

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-13 DOI: 10.1016/j.dam.2024.09.003

A graph $G$ has the strong parity property if for every subset $X \subseteq V (G)$ with $| X |$ even, $G$ has a spanning subgraph $F$ satisfying $δ (F) \geq 1$ , $d_{F} (u) \equiv 1$ (mod 2) for any $u \in X$ , and $d_{F} (v) \equiv 0$ (mod 2) for any $v \in V (G) ∖ X$ . In this paper, we give a spectral radius condition to guarantee that a connected graph has the strong parity property.

如果对于 |X| 偶数的每个子集 X⊆V(G)，G 有一个跨子图 F，满足 δ(F)≥1，对于任意 u∈X 的 dF(u)≡1（mod 2），以及对于任意 v∈V(G)∖X 的 dF(v)≡0（mod 2），则图 G 具有强奇偶性。本文给出了一个谱半径条件，以保证连通图具有强奇偶性。

引用次数: 0

An efficient algorithm for group testing with runlength constraints 具有运行长度限制的分组测试高效算法

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-13 DOI: 10.1016/j.dam.2024.09.001

In this paper, we provide an efficient algorithm to construct almost optimal $(k, n, d)$ -superimposed codes with runlength constraints. A $(k, n, d)$ -superimposed code of length $t$ is a $t \times n$ binary matrix such that any two 1’s in each column are separated by a run of at least $d$ 0’s, and such that for any column $c$ and any other $k - 1$ columns, there exists a row where $c$ has 1 and all the remaining $k - 1$ columns have 0. These combinatorial structures were introduced by Agarwal et al. (2020), in the context of Non-Adaptive Group Testing algorithms with runlength constraints.

By using Moser and Tardos’ constructive version of the Lovász Local Lemma, we provide an efficient randomized Las Vegas algorithm of complexity $Θ (t n^{2})$ for the construction of $(k, n, d)$ -superimposed codes of length $t = O (d k log n + k^{2} log n)$ . We also show that the length of our codes is shorter, for $n$ sufficiently large, than that of the codes whose existence was proved in Agarwal et al. (2020).

在本文中，我们提供了一种高效算法，用于构建具有运行长度限制的几乎最优的（k,n,d）叠加码。长度为 t 的（k,n,d）叠加码是一个 t×n 二进制矩阵，每列中的任意两个 1 之间至少有 d 个 0 隔开，并且对于任意列 c 和任意其他 k-1 列，存在一行 c 为 1，其余 k-1 列均为 0。(通过使用 Moser 和 Tardos 的构造版 Lovász Local Lemma，我们提供了一种复杂度为 Θ(tn2)的高效随机拉斯维加斯算法，用于构建长度为 t=O(dklogn+k2logn) 的 (k,n,d) 叠加码。）我们还证明，在 n 足够大的情况下，我们的代码长度比 Agarwal 等人 (2020) 中证明存在的代码长度更短。

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引用次数: 0

Maximum bisections of graphs without cycles of length four and five 没有长度为四和五的循环的图形的最大平分线

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-13 DOI: 10.1016/j.dam.2024.08.024

A bisection of a graph is a bipartition of its vertex set in which the two parts differ in size by at most 1, and its size is the number of edges which across the two parts. Let $G$ be a graph with $n$ vertices, $m$ edges and degree sequence $d_{1}, d_{2}, \dots, d_{n}$ . Motivated by a few classical results on Max-Cut of graphs, Lin and Zeng proved that if $G$ is ${C_{4}, C_{6}}$ -free and has a perfect matching, then $G$ has a bisection of size at least $m / 2 + Ω (\sum_{i = 1}^{n} \sqrt{d_{i}})$ , and conjectured the same bound holds for $C_{4}$ -free graphs with perfect matchings. In this paper, we confirm the conjecture under the additional condition that $G$ is $C_{5}$ -free.

图的一分为二是其顶点集的两部分，其中两部分的大小最多相差 1，其大小是横跨两部分的边的数量。假设 G 是一个有 n 个顶点、m 条边和阶数序列 d1、d2...、dn 的图。受一些关于图的 Max-Cut 经典结果的启发，Lin 和 Zeng 证明了如果 G 是{C4,C6}-free 并且有一个完全匹配，那么 G 有一个大小至少为 m/2+Ω(∑i=1ndi)的分段。在本文中，我们在 G 无 C5 的附加条件下证实了这一猜想。

引用次数: 0

Two-disjoint-cycle-cover edge/vertex bipancyclicity of star graphs 星形图的两两相接-循环-覆盖边/顶点双周期性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-13 DOI: 10.1016/j.dam.2024.09.004

A bipartite graph $G$ is two-disjoint-cycle-cover edge $[r_{1}, r_{2}]$ -bipancyclic if, for any vertex-disjoint edges $u v$ and $x y$ in $G$ and any even integer $ℓ$ satisfying $r_{1} ⩽ ℓ ⩽ r_{2}$ , there exist vertex-disjoint cycles $C_{1}$ and $C_{2}$ such that $| V (C_{1}) | = ℓ$ , $| V (C_{2}) | = | V (G) | - ℓ$ , $u v \in E (C_{1})$ and $x y \in E (C_{2})$ . In this paper, we prove that the $n$ -star graph $S_{n}$ is two-disjoint-cycle-cover edge $[6, \frac{n!}{2}]$ -bipancyclic for $n ⩾ 5$ , and thus it is two-disjoint-cycle-cover vertex $[6, \frac{n!}{2}]$ -bipancyclic for $n ⩾ 5$ . Additionally, it is examined that $S_{n}$ is two-disjoint-cycle-cover $[6, \frac{n!}{2}]$ -bipancyclic for $n ⩾ 4$ .

如果对于 G 中任意顶点相接的边 uv 和 xy 以及满足 r1⩽ℓ⩽r2.的任意偶整数 ℓ，则双矢点图 G 是双相接循环覆盖边 [r1,r2]-bipancyclic 图、存在顶点相交的循环 C1 和 C2，使得|V(C1)|=ℓ，|V(C2)|=|V(G)|-ℓ，uv∈E(C1) 和 xy∈E(C2) 。本文证明了 n 星图 Sn 在 n⩾5 时是两两相交循环覆盖边 [6,n!2]- 双峰环形，因此在 n⩾5 时是两两相交循环覆盖顶点 [6,n!2]- 双峰环形。另外，检验 Sn 是 n⩾4 的二相交循环顶点 [6,n!2]-双性环。

引用次数: 0

Dips at small sizes for topological graph obstruction sets 拓扑图阻碍集在小尺寸时的衰减

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-12 DOI: 10.1016/j.dam.2024.08.022

The Graph Minor Theorem of Robertson and Seymour implies a finite set of obstructions for any minor closed graph property. We show that there are only three obstructions to knotless embedding of size 23, which is far fewer than the 92 of size 22 and the hundreds known to exist at larger sizes. We describe several other topological properties whose obstruction set demonstrates a similar dip at small size. For order ten graphs, we classify the 35 obstructions to knotless embedding and the 49 maximal knotless graphs.

罗伯逊（Robertson）和西摩（Seymour）的 "图形次要定理"（Graph Minor Theorem of Robertson and Seymour）意味着任何次要封闭图形属性的障碍都是有限的。我们证明，大小为 23 的无结嵌入只有三个障碍，远远少于大小为 22 的 92 个障碍，也少于已知在更大大小时存在的数百个障碍。我们还描述了其他几种拓扑性质，它们的障碍集在小尺寸时也显示出类似的下降趋势。对于十阶图形，我们对 35 个无结嵌入障碍和 49 个最大无结图形进行了分类。

引用次数: 0

Large trees with maximal inverse sum indeg index have no vertices of degree 2 or 3 具有最大逆和 indeg 指数的大树没有阶数为 2 或 3 的顶点

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-10 DOI: 10.1016/j.dam.2024.09.006

The inverse sum indeg ( $I S I$ ) index has attracted more and more attentions, because of its significant applications in chemistry. A basic problem in the study of this topological index is the characterization trees with maximal $I S I$ value. Let $T$ be such a tree of order $n \geq 20$ . Recently, Lin et al. (2022) claimed that $T$ has no vertices of degree 2. However, errors were found in their proofs. Since this result is quite important, we give a correction to the proof. Furthermore, we extend the result by proving that $T$ has no vertices of degree 2 or 3 if $n \geq 58$ .

由于其在化学中的重要应用，逆和指数（ISI）受到越来越多的关注。研究该拓扑指数的一个基本问题是找出 ISI 值最大的树的特征。设 T 是这样一棵阶数 n≥20 的树。最近，Lin 等人（2022 年）声称 T 没有阶数为 2 的顶点。由于这一结果相当重要，我们对证明进行了修正。此外，我们还扩展了这一结果，证明如果 n≥58 则 T 没有阶数为 2 或 3 的顶点。

引用次数: 0

On injective edge-coloring of graphs with maximum degree 4 论最大阶数为 4 的图的注边着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-09-10 DOI: 10.1016/j.dam.2024.09.005

An edge-coloring of a graph $G$ is injective if for any two distinct edges $e_{1}$ and $e_{2}$ , the colors of $e_{1}$ and $e_{2}$ are distinct if they are at distance 2 in $G$ or in a common triangle. The injective chromatic index of $G$ , $χ_{i n j}^{'} (G)$ , is the minimum number of colors needed for an injective edge-coloring of $G$ . In this paper, we prove that if $G$ is graph with $Δ (G) = 4$ and maximum average degree is less than $\frac{5}{2}$ (resp. $\frac{13}{5}$ , $\frac{36}{13}$ ), then $χ_{i n j}^{'} (G) \leq 6$ (resp. 7, 8).

如果对于任意两条不同的边 e1 和 e2，如果 e1 和 e2 在 G 中的距离为 2 或在一个公共三角形中，则它们的颜色是不同的，则图 G 的边着色是注入式的。本文将证明，如果 G 是 Δ(G)=4 的图，且最大平均度小于 52（即 135，3613），则 χinj′(G)≤6（即 7，8）。

引用次数: 0

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