Discrete Applied Mathematics最新文献

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Matchings in bipartite graphs with a given number of cuts 具有给定切分数的二方图中的匹配关系

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-24 DOI: 10.1016/j.dam.2024.08.012

Jinfeng Liu, Fei Huang

A matching in a graph is a set of pairwise nonadjacent edges. Denote by $m (G, k)$ the number of matchings of cardinality $k$ in a graph $G$ . A quasi-order $⪯$ is defined by $G ⪯ H$ whenever $m (G, k) \leq m (H, k)$ holds for all $k$ . Let ${BG}_{1} (n, γ)$ be the set of connected bipartite graphs with $n$ vertices and $γ$ cut vertices, and ${BG}_{2} (n, γ)$ be the set of connected bipartite graphs with $n$ vertices and $γ$ cut edges. We determine the greatest and least elements with respect to this quasi-order in ${BG}_{1} (n, γ)$ and the greatest element in ${BG}_{2} (n, γ)$ for all values of $n$ and $γ$ . As corollaries, we find that these graphs maximize (resp. minimize) the Hosoya index and the matching energy within the respective sets.

图中的匹配是指成对不相邻边的集合。当 m(G,k)≤m(H,k)对所有 k 都成立时，准序⪯的定义是 G⪯H。让 BG1(n,γ)是具有 n 个顶点和 γ 个切顶的连通双方图的集合，BG2(n,γ)是具有 n 个顶点和 γ 条切边的连通双方图的集合。我们确定了在所有 n 和 γ 值下，BG1(n,γ) 中与此准序相关的最大元素和最小元素，以及 BG2(n,γ) 中的最大元素。

引用次数: 0

Diverse fair allocations: Complexity and algorithms 多样化的公平分配：复杂性与算法

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-24 DOI: 10.1016/j.dam.2024.07.045

Harshil Mittal, Saraswati Nanoti, Aditi Sethia

In this work, we initiate the study of diversity of solutions in the context of fair division of indivisible goods. In particular, we explore the notions of disjoint, distinct and symmetric allocations and study their complexity in terms of the fairness notions of envy-freeness and equitability up to one item. We show that for binary valuations, the above problems are polynomial time solvable. In contrast we show NP-hardness of disjoint and symmetric case, when the valuations are additive.

在这项工作中，我们开始研究不可分割物品公平分配中解决方案的多样性。特别是，我们探讨了不相交、不同和对称分配的概念，并从无嫉妒和公平性概念的角度研究了它们的复杂性。我们证明，对于二进制估值，上述问题都可以在多项式时间内解决。相反，当估值为加法时，我们证明了不相交和对称情况下的 NP 难度。

引用次数: 0

On a conjecture of TxGraffiti: Relating zero forcing and vertex covers in graphs 关于 TxGraffiti 的一个猜想：图中零强迫与顶点覆盖的关系

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-22 DOI: 10.1016/j.dam.2024.08.006

Boris Brimkov , Randy Davila , Houston Schuerger , Michael Young

In this paper, we showcase the process of using an automated conjecturing program called TxGraffiti written and maintained by the second author. We begin by proving a conjecture formulated by TxGraffiti that for a claw-free graph $G$ , the vertex cover number $β (G)$ is greater than or equal to the zero forcing number $Z (G)$ . Our proof of this result is constructive, and yields a polynomial time algorithm to find a zero forcing set with cardinality $β (G)$ . We also use the output of TxGraffiti to construct several infinite families of claw-free graphs for which $Z (G) = β (G)$ . Additionally, inspired by the aforementioned conjecture of TxGraffiti, we also prove a more general relationship between the zero forcing number and the vertex cover number for any connected graph with maximum degree $Δ \geq 3$ , namely that $Z (G) \leq (Δ - 2) β (G)$ +1.

在本文中，我们展示了由第二作者编写和维护的名为 TxGraffiti 的自动猜想程序的使用过程。我们首先证明了 TxGraffiti 提出的一个猜想：对于无爪图 G，顶点覆盖数 β(G) 大于或等于零强制数 Z(G)。我们对这一结果的证明是构造性的，并产生了一种多项式时间算法，可以找到具有心数 β(G) 的零强制集。我们还利用 TxGraffiti 的输出构建了 Z(G)=β(G)的多个无爪图无限族。此外，受 TxGraffiti 上述猜想的启发，我们还证明了任何最大阶数 Δ≥3 的连通图的零强制数和顶点覆盖数之间更一般的关系，即 Z(G)≤(Δ-2)β(G)+1 。

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引用次数: 0

Algorithmic results for weak Roman domination problem in graphs 图中弱罗马支配问题的算法结果

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-22 DOI: 10.1016/j.dam.2024.08.007

Kaustav Paul, Ankit Sharma, Arti Pandey

<div>Consider a graph <math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math> and a function <math><mrow><mi>f</mi><mo>:</mo><mi>V</mi><mo>→</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow></math>. A vertex <math><mi>u</mi></math> with <math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math> is defined as undefended by <math><mi>f</mi></math> if it lacks adjacency to any vertex with a positive <math><mi>f</mi></math>-value. The function <math><mi>f</mi></math> is said to be a weak Roman dominating function (WRD function) if, for every vertex <math><mi>u</mi></math> with <math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mrow></math>, there exists a neighbor <math><mi>v</mi></math> of <math><mi>u</mi></math> with <math><mrow><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>></mo><mn>0</mn></mrow></math> and a new function <math><mrow><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>:</mo><mi>V</mi><mo>→</mo><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>}</mo></mrow></mrow></math> defined in the following way: <math><mrow><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>=</mo><mn>1</mn></mrow></math>, <math><mrow><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math>, and <math><mrow><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow></mrow></math>, for all vertices <math><mi>w</mi></math> in <math><mrow><mi>V</mi><mo>∖</mo><mrow><mo>{</mo><mi>u</mi><mo>,</mo><mi>v</mi><mo>}</mo></mrow></mrow></math>; so that no vertices are undefended by <math><msup><mrow><mi>f</mi></mrow><mrow><mo>′</mo></mrow></msup></math>. The total weight of <math><mi>f</mi></math> is equal to <math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>v</mi><mo>∈</mo><mi>V</mi></mrow></msub><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math>, and is denoted as <math><mrow><mi>w</mi><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></math>. The Weak Roman domination number denoted by <math><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>r</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>, represents <math><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo><mi>w</mi><mrow><mo>(</mo><m

考虑图 G=(V,E)和函数 f:V→{0,1,2}。如果 f(u)=0 的顶点 u 与任何 f 值为正的顶点都不相邻，则定义为 f 的不防御顶点。如果对于每个 f(u)=0 的顶点 u，都存在一个 f(v)>0 的邻接点 v 和一个新函数 f′，则称函数 f 为弱罗马支配函数（WRD 函数）：对于 V∖{u,v} 中的所有顶点 w，f′(u)=1，f′(v)=f(v)-1，f′(w)=f(w)；因此没有顶点不被 f′ 防御。f 的总权重等于∑v∈Vf(v)，记为 w(f)。弱罗马支配数用 γr(G)表示，表示最小{w(f)|f 是 G 的 WRD 函数}。对于给定的图 G，寻找权重为 γr(G)的 WRD 函数的问题被定义为最小弱罗马支配数问题。众所周知，这个问题对于双向图和弦图来说是 NP 难的。本文将进一步研究该问题的算法复杂性。我们证明了星凸双artite图和梳状双artite图的 NP 难性，它们都是双artite图的子类。此外，我们还证明了对于有界度星凸双态图，该问题是有效可解的。我们还证明了分裂图（弦图的一个子类）问题的 NP 难度。从积极的一面来看，我们提出了一种多项式时间算法来解决 P4 稀疏图的问题。此外，我们还提出了一些近似结果。

引用次数: 0

A characterization of graphs whose vertex set can be partitioned into a total dominating set and an independent dominating set 顶点集可划分为总支配集和独立支配集的图的特征描述

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-22 DOI: 10.1016/j.dam.2024.08.008

Teresa W. Haynes , Michael A. Henning

A set $S$ of vertices in a graph $G$ is a dominating set if every vertex not in $S$ is adjacent to a vertex in $S$ . If, in addition, $S$ is an independent set, then $S$ is an independent dominating set, while if every vertex of the dominating set $S$ is adjacent to a vertex in $S$ , then $S$ is a total dominating set of $G$ . A fundamental problem in domination theory in graphs is to determine which graphs have the property that their vertex set can be partitioned into two types of dominating sets. In particular, we are interested in graphs whose vertex set can be partitioned into a total dominating set and an independent dominating set. In this paper, we solve this problem by providing a constructive characterization of such graphs. We show that all such graphs can be constructed starting from two base graphs and applying a sequence of eleven operations.

如果不在 S 中的每个顶点都与 S 中的顶点相邻，那么图 G 中的顶点集 S 就是支配集。此外，如果 S 是独立集，那么 S 就是独立支配集；如果支配集 S 中的每个顶点都与 S 中的顶点相邻，那么 S 就是 G 的总支配集。特别是，我们对顶点集可划分为总支配集和独立支配集的图感兴趣。在本文中，我们通过提供此类图的构造性特征来解决这个问题。我们证明，所有此类图都可以从两个基图开始，应用 11 个操作序列来构造。

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引用次数: 0

Maxima of the Q-index for 3K3-free graphs 无 3K3 图形的 Q 指数最大值

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-21 DOI: 10.1016/j.dam.2024.08.004

Yanting Zhang, Ligong Wang

The $Q$ -index of a graph $G$ is the largest eigenvalue of its $Q$ -matrix $Q (G) = D (G) + A (G)$ , where $D (G)$ and $A (G)$ are the diagonal matrix of vertex degrees and the adjacency matrix of $G$ , respectively. Let $3 K_{3}$ denote the graph consisting of three vertex-disjoint triangles. A graph is called $3 K_{3}$ -free if it does not contain $3 K_{3}$ as a subgraph. In this paper, we present a sharp upper bound on the $Q$ -index of $3 K_{3}$ -free graphs of order $n \geq 453$ , and characterize the unique extremal graph which attains the bound.

图 G 的 Q 指数是其 Q 矩阵 Q(G)=D(G)+A(G) 的最大特征值，其中 D(G) 和 A(G) 分别是顶点度对角矩阵和 G 的邻接矩阵。让 3K3 表示由三个顶点相交的三角形组成的图。如果一个图的子图中不包含 3K3，则该图被称为无 3K3。本文提出了阶数 n≥453 的无 3K3 图的 Q 指数的尖锐上限，并描述了达到该上限的唯一极值图的特征。

引用次数: 0

Packing coloring of hypercubes with extended Hamming codes 用扩展汉明码对超立方体进行打包着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-20 DOI: 10.1016/j.dam.2024.07.048

Petr Gregor , Jaka Kranjc , Borut Lužar , Kenny Štorgel

A packing $k$ -coloring of a graph $G$ is a mapping assigning a positive integer (a color) from the set $(1, \dots, k)$ to every vertex of $G$ such that every two distinct vertices of color $c$ are at distance at least $c + 1$ . The minimum value $k$ such that $G$ admits a packing $k$ -coloring is called the packing chromatic number of $G$ . In this paper, we continue the study of the packing chromatic number of hypercubes and we improve the upper bounds reported by Torres and Valencia-Pabon (2015) by presenting recursive constructions of subsets of distant vertices making use of the properties of the extended Hamming codes. We also answer in negative a question on the packing chromatic number of Cartesian products raised by Brešar et al. (2007).

图 G 的打包 k 着色是给 G 的每个顶点从集合 1，...，k 中指定一个正整数（一种颜色）的映射，使得颜色 c 的每两个不同顶点的距离至少为 c+1。本文将继续研究超立方体的堆积色度数，并利用扩展汉明码的特性，提出了远距离顶点子集的递归构造，从而改进了 Torres 和 Valencia-Pabon (2015) 报告的上界。我们还否定地回答了 Brešar 等人（2007 年）提出的关于笛卡尔积的包装色度数的问题。

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引用次数: 0

Hamiltonicity of covering graphs of trees 树覆盖图的哈密顿性

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-17 DOI: 10.1016/j.dam.2024.07.044

Peter Bradshaw , Zhilin Ge , Ladislav Stacho

In this paper, we consider covering graphs obtained by lifting a tree with a loop at each vertex as a voltage graph over a cyclic group. We generalize a tool of Hell et al. (2020), known as the billiard strategy, for constructing Hamiltonian cycles in the covering graphs of paths. We show that our extended tool can be used to provide new sufficient conditions for the Hamiltonicity of covering graphs of trees that are similar to those of Batagelj and Pisanski (1982) and of Hell et al. (2020). Next, we focus specifically on covering graphs obtained from trees lifted as voltage graphs over cyclic groups $Z_{p}$ of large prime order $p$ . We prove that for a given reflexive tree $T$ whose edge labels are assigned uniformly at random from a finite set, the corresponding lift is almost surely Hamiltonian for a large enough prime-ordered cyclic group $Z_{p}$ . Finally, we show that if a reflexive tree $T$ is lifted over a group $Z_{p}$ of a large prime order, then for any assignment of nonzero elements of $Z_{p}$ to the edges of $T$ , the corresponding cover of $T$ has a large circumference.

在本文中，我们考虑了通过将每个顶点都有一个循环的树提升为循环群上的电压图而得到的覆盖图。我们推广了 Hell 等人（2020）的一种工具，即台球策略，用于构建路径覆盖图中的哈密尔顿循环。我们证明，我们的扩展工具可以用来为树的覆盖图的汉密尔顿性提供新的充分条件，这些条件与 Batagelj 和 Pisanski (1982) 以及 Hell 等人 (2020) 的条件相似。我们证明，对于边标签从有限集合中均匀随机分配的给定反折树 T，对于足够大的素序循环群 Zp，相应的提升几乎肯定是哈密顿性的。最后，我们证明，如果一棵反折树 T 被提升到一个大素数阶的群 Zp 上，那么对于将 Zp 的非零元素分配给 T 的边的任何赋值，T 的相应覆盖都有一个大周长。

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引用次数: 0

On b-greedy colourings and z-colourings 关于b-贪婪着色和z-着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-17 DOI: 10.1016/j.dam.2024.08.001

Jonas Costa Ferreira da Silva , Frédéric Havet

A b-greedy colouring is a colouring which is both a b-colouring and a greedy colouring. A z-colouring is a b-greedy colouring such that a b-vertex of the largest colour is adjacent to a b-vertex of every other colour. The b-Grundy number (resp. z-number) of a graph is the maximum number of colours in a b-greedy colouring (resp. z-colouring) of it. In this paper, we study those two parameters. We show that similarly to the z-number, the b-Grundy number is not monotone and can be arbitrarily smaller than the minimum of the Grundy number and the b-chromatic number. We also describe a polynomial-time algorithm that decides whether a given $k$ -regular graph has b-Grundy number (resp. z-number) equal to $k + 1$ . We also prove that every cubic graph with no induced 4-cycle has b-Grundy number and z-number exactly 4.

b-贪婪着色是一种既是 b-着色又是贪婪着色的着色。z-着色是一种 b-贪婪着色，即最大颜色的 b 顶点与其他颜色的 b 顶点相邻。图形的 b 格伦迪数（或 z 数）是图形的 b 贪婪着色（或 z 着色）中的最大颜色数。本文将研究这两个参数。我们证明，与 z 数类似，b-格兰迪数也不是单调的，可以任意小于格兰迪数和 b-色数的最小值。我们还描述了一种多项式时间算法，它可以判定给定的 k 个规则图的 b 格兰迪数（或 z 数）是否等于 k+1。我们还证明了每一个没有诱导 4 循环的立方图的 b-Grundy 数和 z 数恰好都是 4。

引用次数: 0

On locally identifying coloring of Cartesian product and tensor product of graphs 论图的笛卡尔积和张量积的局部标识着色

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics

Pub Date : 2024-08-17 DOI: 10.1016/j.dam.2024.07.046

Sriram Bhyravarapu , Swati Kumari , I. Vinod Reddy

<div>For a positive integer <math><mi>k</mi></math>, a proper <math><mi>k</mi></math>-coloring of a graph <math><mi>G</mi></math> is a mapping <math><mrow><mi>f</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><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></mrow></mrow></math> such that <math><mrow><mi>f</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≠</mo><mi>f</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math> for each edge <math><mrow><mi>u</mi><mi>v</mi></mrow></math> of <math><mi>G</mi></math>. The smallest integer <math><mi>k</mi></math> for which there is a proper <math><mi>k</mi></math>-coloring of <math><mi>G</mi></math> is called the chromatic number of <math><mi>G</mi></math>, denoted by <math><mrow><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>. A locally identifying coloring (for short, lid-coloring) of a graph <math><mi>G</mi></math> is a proper <math><mi>k</mi></math>-coloring of <math><mi>G</mi></math> such that every pair of adjacent vertices with distinct closed neighborhoods has distinct set of colors in their closed neighborhoods. The smallest integer <math><mi>k</mi></math> such that <math><mi>G</mi></math> has a lid-coloring with <math><mi>k</mi></math> colors is called locally identifying chromatic number (for short, lid-chromatic number) of <math><mi>G</mi></math>, denoted by <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>l</mi><mi>i</mi><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math>.This paper studies the lid-coloring of the Cartesian product and tensor product of two graphs. We prove that if <math><mi>G</mi></math> and <math><mi>H</mi></math> are two connected graphs having at least two vertices, then (a) <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>l</mi><mi>i</mi><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>□</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mi>χ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow></math> and (b) <math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>l</mi><mi>i</mi><mi>d</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>×</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mi>χ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mi>χ</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow></mrow></math>. Here <math><mrow><mi>G</mi><mo>□</mo><mi>H</mi></mrow></math> and <math><mrow><mi>G</mi><mo>×</mo><mi>H</mi></mrow></math> denote the Cartesian and tensor products of <math><mi>G</mi></math> and

对于正整数 k，图 G 的适当 k 着色是一个映射 f:V(G)→{1,2,...,k}，使得对于 G 的每条边 uv，f(u)≠f(v)。G 存在适当 k 着色的最小整数 k 称为 G 的色度数，用 χ(G) 表示。图 G 的局部标识着色（简称为盖子着色）是 G 的适当 k 着色，使得每一对具有不同封闭邻域的相邻顶点在其封闭邻域中都有一组不同的颜色。使 G 具有 k 种颜色的顶点着色的最小整数 k 称为 G 的局部标识色度数（简称顶点着色数），用 χlid(G)表示。我们证明，如果 G 和 H 是至少有两个顶点的两个连通图，那么 (a) χlid(G□H)≤χ(G)χ(H)-1；(b) χlid(G×H)≤χ(G)χ(H)。这里，G□H 和 G×H 分别表示 G 和 H 的笛卡尔积和张量积。我们确定了 Cm□Pn、Cm□Cn、Pm×Pn、Cm×Pn 和 Cm×Cn 的立德数，其中 Cm 和 Pn 分别表示 m 个顶点上的循环和 n 个顶点上的路径。

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