Pub Date : 2021-03-17DOI: 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.
{"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}
Pub Date : 2021-02-25DOI: 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.
{"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}
Pub Date : 2021-01-01DOI: 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.
{"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}
Pub Date : 2021-01-01DOI: 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.
{"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}
Pub Date : 2021-01-01DOI: 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.
{"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}
Pub Date : 2021-01-01DOI: 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.
{"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}
Pub Date : 2020-09-17DOI: 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.
{"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}
Pub Date : 2020-07-13DOI: 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 $mathscr{C}_1$.
{"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}
Pub Date : 2020-01-01DOI: 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}
Pub Date : 2020-01-01DOI: 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(∆)。
{"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}
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