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Ultrametrics and Complete Multipartite Graphs 超度量与完全多部图
Q4 Mathematics Pub Date : 2021-03-17 DOI: 10.20429/tag.2022.090108
V. Bilet, O. Dovgoshey, Yu. M. Kononov
We describe the class of graphs for which all metric spaces with diametrical graphs belonging to this class are ultrametric. It is shown that a metric space (X, d) is ultrametric iff the diametrical graph of the metric dε(x, y) = max{d(x, y), ε} is either empty or complete multipartite for every ε > 0. A refinement of the last result is obtained for totally bounded spaces. Moreover, using complete multipartite graphs we characterize the compact ultrametrizable topological spaces. The bounded ultrametric spaces, which are weakly similar to unbounded ones, are also characterized via complete multipartite graphs.
我们描述了一类图,对于这类图,具有直径图的所有度量空间都是超度量的。证明了度量空间(X,d)是超度量的,当度量d的直径图ε(X,y)=max{d(X,y),ε}对于每个ε>0为空或完全多部分时。对于完全有界空间,得到了最后一个结果的精化。此外,利用完全多部分图刻画了紧致超可度量拓扑空间。与无界空间弱相似的有界超度量空间也通过完全多部分图来刻画。
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引用次数: 4
Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets 闭凸集诱导的单位距离子图的连通性
Q4 Mathematics Pub Date : 2021-02-25 DOI: 10.20429/tag.2022.090102
Remie Janssen, L. V. Steijn
The unit distance graph $G_{mathbb{R}^d}^1$ is the infinite graph whose nodes are points in $mathbb{R}^d$, with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version $G_{mathbb{R}^2}^1$ of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of $G_{mathbb{R}^d}^1$ to closed convex subsets $X$ of $mathbb{R}^d$. We show that the graph $G_{mathbb{R}^d}^1[X]$ is connected precisely when the radius of $r(X)$ of $X$ is equal to 0, or when $r(X)geq 1$ and the affine dimension of $X$ is at least 2. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1.
单位距离图$G_{mathbb{R}^d}^1$是无限图,其节点是$mathbb{R}^ d$中的点,如果两点之间的欧几里得距离为1,则两点之间有边。该图的2维版本$G_{mathbb{R}^2}^1$通常被研究其色数,如在Hadwiger-Nelson问题中。然而,单位距离图的其他性质很少被研究。这里,我们考虑$G_{mathbb{R}^d}^1$对$mathbb{R}^ d$的闭凸子集$X$的限制。我们证明了当$X$的$R(X)$半径等于0时,或者当$R(X)geq1$和$X$仿射维数至少为2时,图$G_{mathbb{R}^d}^1[X]$是精确连接的。对于超矩形,在半径恰好为1的临界情况下,我们给出了图直径的边界。
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引用次数: 0
The Conditional Strong Matching Preclusion of Augmented Cubes 增广立方体的条件强匹配排除
Q4 Mathematics Pub Date : 2021-01-01 DOI: 10.20429/TAG.2021.080105
Mohamad Abdallah, E. Cheng
The strong matching preclusion is a measure for the robustness of interconnection networks in the presence of node and/or link failures. However, in the case of random link and/or node failures, it is unlikely to find all the faults incident and/or adjacent to the same vertex. This motivates Park et al. to introduce the conditional strong matching preclusion of a graph. In this paper we consider the conditional strong matching preclusion problem of the augmented cube AQn, which is a variation of the hypercube Qn that possesses favorable properties.
强匹配排除是在存在节点和/或链路故障的情况下互连网络鲁棒性的度量。然而,在随机链路和/或节点故障的情况下,不太可能找到所有故障事件和/或邻近同一顶点。这促使Park等人引入图的条件强匹配排除。本文考虑增广立方Qn的条件强匹配排除问题,增广立方Qn是超立方Qn的一种变体,具有良好的性质。
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引用次数: 1
On delta^(k)-colouring of Powers of Paths and Cycles 关于^(k)-路径和环的幂的着色
Q4 Mathematics Pub Date : 2021-01-01 DOI: 10.20429/tag.2021.080203
Merlin Thomas Ellumkalayil, S. Naduvath
In an improper vertex colouring of a graph, adjacent vertices are permitted to receive same colours. An edge of an improperly coloured graph is said to be a bad edge if its end vertices have the same colour. A near-proper colouring of a graph is a colouring which minimises the number of bad edges by restricting the number of colour classes that can have adjacency among their own elements. The δ(k)colouring is a near-proper colouring of G consisting of k given colours, where 1 ≤ k ≤ χ(G)− 1, which minimises the number of bad edges by permitting at most one colour class to have adjacency among the vertices in it. In this paper, we discuss the number of bad edges of powers of paths and cycles.
在不适当的顶点着色的图中,相邻的顶点被允许获得相同的颜色。如果颜色不正确的图的端点颜色相同,则称其为坏边。图的近似适当着色是一种着色,它通过限制可以在它们自己的元素之间具有邻接的颜色类的数量来最小化坏边的数量。δ(k)着色是由k个给定颜色组成的G的近似适当着色,其中1≤k≤χ(G)−1,通过允许最多一个颜色类中的顶点之间具有邻接性来最小化坏边的数量。本文讨论了路径和环的幂坏边的个数。
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引用次数: 0
Skolem Number of Cycles and Grid Graphs Skolem循环数和网格图
Q4 Mathematics Pub Date : 2021-01-01 DOI: 10.20429/tag.2021.080204
B. Carrigan, J. Asplund
A Skolem sequence can be thought of as a labelled path where two vertices with the same label are that distance apart. This concept has naturally been generalized to labellings of other graphs, but always using at most two of any integer label. Given that more than two vertices can be mutually distance d apart, we define a new generalization of a Skolem sequence on graphs that we call a proper Skolem labelling. This brings rise to the question; “what is the smallest set of consecutive positive integers we can use to proper Skolem label a graph?” This will be known as the Skolem number of the graph. In this paper we give the Skolem number for cycles and grid graphs, while also providing other related results along the way.
Skolem序列可以被认为是一个有标签的路径,其中两个具有相同标签的顶点之间的距离是相同的。这个概念自然被推广到其他图的标记,但总是最多使用两个整数标记。假设两个以上的顶点之间的相互距离为d,我们定义了图上Skolem序列的一种新的泛化,我们称之为适当的Skolem标记。这就引出了一个问题;"我们可以用来对图进行适当Skolem标记的连续正整数的最小集合是多少? "这将被称为图的Skolem数。本文给出了循环和网格图的Skolem数,同时也给出了其他相关的结果。
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引用次数: 1
The Structure Of Functional Graphs For Functions From A Finite Domain To Itself For Which A Half Iterate Exists 从有限域到自身且存在半迭代函数的泛函图结构
Q4 Mathematics Pub Date : 2021-01-01 DOI: 10.20429/tag.2021.080108
P. Kozyra
The notion of a replica of a nontrivial in-tree is defined. A result enabling to determine whether an in-tree is a replica of another in-tree employing an injective mapping between some subsets of sources of these in-trees is presented. There are given necessary and sufficient conditions for the existence of a functional square root of a function from a finite set to itself through presenting necessary and sufficient conditions for the existence of a square root of a component of the functional graph for the function and for the existence of a square root of the union of two components of the functional graph for the function containing cycles of the same length using the concept of the replica.
定义了非平凡in-tree副本的概念。给出了一个结果,该结果可以利用这些内树的源的某些子集之间的内射映射来确定一个内树是否是另一个内树的副本。利用复形的概念,给出了函数从有限集合到自身的函数的泛函平方根存在的充分必要条件,并给出了该函数的泛函图分量的平方根存在的充分必要条件,以及包含相同长度循环的函数的泛函图两个分量的并并的平方根存在的充分必要条件。
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引用次数: 0
Radio Number of Hamming Graphs of Diameter 3 直径为3的Hamming图的无线电数
Q4 Mathematics Pub Date : 2020-09-17 DOI: 10.20429/tag.2022.090210
Jason DeVito, Amanda Niedzialomski, J. Warren
For $G$ a simple, connected graph, a vertex labeling $f:V(G)rightarrow mathbb{Z}_+$ is called a $textit{radio labeling of}$ $G$ if it satisfies $|f(u)-f(v)|geq operatorname{diam}(G) + 1 - d(u,v)$ for all distinct vertices $u,vin V(G)$. The $textit{radio number}$ of $G$ is the minimal span over all radio labelings of $G$. If a bijective radio labeling onto ${1,2,...,|V(G)|}$ exists, $G$ is called a $textit{radio graceful graph}$. We determine the radio number of all diameter $3$ Hamming graphs and show that an infinite subset of them is radio graceful.
对于$G$一个简单的连通图,一个标记为$f:V(G)rightarrowmathbb的顶点{Z}_+$称为$$G$的$textit{无线电标记},如果它满足v(G)$中所有不同顶点$u,v的$|f(u)-f(v)|geq运算符名称{diam}(G)+1-d(u,v)$。$G$的$textit{无线电号码}$是$G$所有无线电标签的最小跨度。如果${1,2,…,|V(G)|}$上存在双射无线电标记,则$G$称为$textit{无线电优美图}$。我们确定了所有直径$3$Hamming图的无线电数,并证明了它们的无限子集是无线电优美的。
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引用次数: 0
A Complete Characterisation of Vertex-multiplications of Trees with Diameter 5 直径为5的树顶点乘法的完整表征
Q4 Mathematics Pub Date : 2020-07-13 DOI: 10.20429/tag.2021.080206
W. Wong, E. G. Tay
Koh and Tay introduced a new family of graphs, $G$ vertex-multiplications, as an extension of complete $n$-partite graphs. They proved a fundamental classification of $G$ vertex-multiplications into three classes $mathscr{C}_0, mathscr{C}_1$ and $mathscr{C}_2$. It was shown that any vertex-multiplication of a tree with diameter at least 3 does not belong to the class $mathscr{C}_2$. Furthermore, for vertex-multiplications of trees with diameter $5$, some necessary and sufficient conditions for $mathscr{C}_0$ were established. In this paper, we give a complete characterisation of vertex-multiplications of trees with diameter $5$ in $mathscr{C}_0$ and $mathscr{C}_1$.
Koh和Tay引入了一个新的图族$G$顶点乘法,作为完整$n$部分图的扩展。他们证明了$G$顶点乘法的基本分类为三个类$mathscr{C}_0,mathscr{C}_1$和$mathscr{C}_2$。结果表明,直径至少为3的树的任何顶点乘法都不属于$mathscr类{C}_2$。此外,对于直径为$5$的树的顶点乘法,$mathscr的一些充要条件{C}_0建立了$。在本文中,我们给出了$mathscr中直径为$5$的树的顶点乘法的一个完整刻画{C}_0$和$mathscr{C}_1$。
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引用次数: 2
An Improvement in the Two-packing Bound Related to Vizing's Conjecture 与维津猜想有关的双填充界的改进
Q4 Mathematics Pub Date : 2020-01-01 DOI: 10.20429/tag.2020.070105
Kimber Wolff
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引用次数: 0
Fault diagnosability of regular graphs 正则图的故障可诊断性
Q4 Mathematics Pub Date : 2020-01-01 DOI: 10.20429/tag.2020.070204
Mei-Mei Gu, Law, Rongxia Hao, E. Cheng
An interconnection network’s diagnosability is an important measure of its selfdiagnostic capability. In 2012, Peng et al. proposed a measure for fault diagnosis of the network, namely, the h-good-neighbor conditional diagnosability, which requires that every fault-free node has at least h fault-free neighbors. There are two well-known diagnostic models, PMC model and MM* model. The h-goodneighbor diagnosability under the PMC (resp. MM*) model of a graph G, denoted by tPMC h (G) (resp. t MM∗ h (G)), is the maximum value of t such that G is h-good-neighbor t-diagnosable under the PMC (resp. MM*) model. In this paper, we study the 2-good-neighbor diagnosability of some general k-regular kconnected graphs G under the PMC model and the MM* model. The main result tPMC 2 (G) = t MM∗ 2 (G) = g(k − 1)− 1 with some acceptable conditions is obtained, where g is the girth of G. Furthermore, the following new results under the two models are obtained: tPMC 2 (HSn) = t MM∗ 2 (HSn) = 4n− 5 for the hierarchical star network HSn, t PMC 2 (S 2 n) = t MM∗ 2 (S 2 n) = 6n− 13 for the split-star networks S2 n and tPMC 2 (Γn(∆)) = t MM∗ 2 (Γn(∆)) = 6n − 16 for the Cayley graph generated by the 2-tree Γn(∆).
互联网络的可诊断性是衡量互联网络自诊断能力的重要指标。2012年,Peng等人提出了一种网络故障诊断的度量,即h-好邻居条件可诊断性,它要求每个无故障节点至少有h个无故障邻居。有两种比较知名的诊断模型:PMC模型和MM*模型。PMC下的h-近邻可诊断性。图G的MM*)模型,用tPMC h (G)表示。t MM * h (G))是t的最大值,使得G在PMC (resp。毫米*)模型。本文研究了一类一般k-正则k连通图G在PMC模型和MM*模型下的2近邻可诊断性。主要结果tPMC 2 (G) = t MM∗2 G (G) = (k−1)−1与一些可接受的条件,在G的周长是G .此外,获得以下新结果在两个模型:tPMC 2(小企业)= t MM∗2(小企业)= 4 n−5等级的星形网络HSn、t PMC 2 (2 n) = t MM∗2 (2 n) = 6 n−13 split-star网络S2 n和tPMC 2(Γn(∆))= t MM∗2(Γn(∆))= 6 n−16凯莱图生成的2-treeΓn(∆)。
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引用次数: 4
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Theory and Applications of Graphs
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