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Theory and Applications of Graphs最新文献

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Ultrametrics and Complete Multipartite Graphs 超度量与完全多部图
Theory and Applications of Graphs Pub Date : 2021-03-17 DOI: 10.20429/tag.2022.090108
V. Bilet, O. Dovgoshey, Yu. M. Kononov
{"title":"Ultrametrics and Complete Multipartite Graphs","authors":"V. Bilet, O. Dovgoshey, Yu. M. Kononov","doi":"10.20429/tag.2022.090108","DOIUrl":"https://doi.org/10.20429/tag.2022.090108","url":null,"abstract":"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.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42610046","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Theory and Applications of Graphs Pub Date : 2021-02-25 DOI: 10.20429/tag.2022.090102
Remie Janssen, L. V. Steijn
{"title":"Connectedness of Unit Distance Subgraphs Induced by Closed Convex Sets","authors":"Remie Janssen, L. V. Steijn","doi":"10.20429/tag.2022.090102","DOIUrl":"https://doi.org/10.20429/tag.2022.090102","url":null,"abstract":"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.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-02-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49239454","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Theory and Applications of Graphs Pub Date : 2021-01-01 DOI: 10.20429/TAG.2021.080105
Mohamad Abdallah, E. Cheng
{"title":"The Conditional Strong Matching Preclusion of Augmented Cubes","authors":"Mohamad Abdallah, E. Cheng","doi":"10.20429/TAG.2021.080105","DOIUrl":"https://doi.org/10.20429/TAG.2021.080105","url":null,"abstract":"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.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Theory and Applications of Graphs Pub Date : 2021-01-01 DOI: 10.20429/tag.2021.080203
Merlin Thomas Ellumkalayil, S. Naduvath
{"title":"On delta^(k)-colouring of Powers of Paths and Cycles","authors":"Merlin Thomas Ellumkalayil, S. Naduvath","doi":"10.20429/tag.2021.080203","DOIUrl":"https://doi.org/10.20429/tag.2021.080203","url":null,"abstract":"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.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545620","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Skolem Number of Cycles and Grid Graphs Skolem循环数和网格图
Theory and Applications of Graphs Pub Date : 2021-01-01 DOI: 10.20429/tag.2021.080204
B. Carrigan, J. Asplund
{"title":"Skolem Number of Cycles and Grid Graphs","authors":"B. Carrigan, J. Asplund","doi":"10.20429/tag.2021.080204","DOIUrl":"https://doi.org/10.20429/tag.2021.080204","url":null,"abstract":"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.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545629","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Theory and Applications of Graphs Pub Date : 2021-01-01 DOI: 10.20429/tag.2021.080108
P. Kozyra
{"title":"The Structure Of Functional Graphs For Functions From A Finite Domain To Itself For Which A Half Iterate Exists","authors":"P. Kozyra","doi":"10.20429/tag.2021.080108","DOIUrl":"https://doi.org/10.20429/tag.2021.080108","url":null,"abstract":"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.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"46 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545542","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Theory and Applications of Graphs Pub Date : 2020-09-17 DOI: 10.20429/tag.2022.090210
Jason DeVito, Amanda Niedzialomski, J. Warren
{"title":"Radio Number of Hamming Graphs of Diameter 3","authors":"Jason DeVito, Amanda Niedzialomski, J. Warren","doi":"10.20429/tag.2022.090210","DOIUrl":"https://doi.org/10.20429/tag.2022.090210","url":null,"abstract":"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.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48207362","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Theory and Applications of Graphs Pub Date : 2020-07-13 DOI: 10.20429/tag.2021.080206
W. Wong, E. G. Tay
{"title":"A Complete Characterisation of Vertex-multiplications of Trees with Diameter 5","authors":"W. Wong, E. G. Tay","doi":"10.20429/tag.2021.080206","DOIUrl":"https://doi.org/10.20429/tag.2021.080206","url":null,"abstract":"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 $mathscr{C}_1$.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43067919","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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
Theory and Applications of Graphs Pub Date : 2020-01-01 DOI: 10.20429/tag.2020.070105
Kimber Wolff
{"title":"An Improvement in the Two-packing Bound Related to Vizing's Conjecture","authors":"Kimber Wolff","doi":"10.20429/tag.2020.070105","DOIUrl":"https://doi.org/10.20429/tag.2020.070105","url":null,"abstract":"","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fault diagnosability of regular graphs 正则图的故障可诊断性
Theory and Applications of Graphs Pub Date : 2020-01-01 DOI: 10.20429/tag.2020.070204
Mei-Mei Gu, Law, Rongxia Hao, E. Cheng
{"title":"Fault diagnosability of regular graphs","authors":"Mei-Mei Gu, Law, Rongxia Hao, E. Cheng","doi":"10.20429/tag.2020.070204","DOIUrl":"https://doi.org/10.20429/tag.2020.070204","url":null,"abstract":"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(∆).","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"67545800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
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