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Theory and Applications of Graphs最新文献

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On the Total Set Chromatic Number of Graphs 图的总色数集
Theory and Applications of Graphs Pub Date : 2022-07-18 DOI: 10.20429/tag.2022.090205
M. A. Tolentino, Gerone Russel Eugenio, M. Ruiz
{"title":"On the Total Set Chromatic Number of Graphs","authors":"M. A. Tolentino, Gerone Russel Eugenio, M. Ruiz","doi":"10.20429/tag.2022.090205","DOIUrl":"https://doi.org/10.20429/tag.2022.090205","url":null,"abstract":"Given a vertex coloring c of a graph, the neighborhood color set of a vertex is defined to be the set of all of its neighbors’ colors. The coloring c is called a set coloring if any two adjacent vertices have different neighborhood color sets. The set chromatic number χ s ( G ) of a graph G is the minimum number of colors required in a set coloring of G . In this work, we investigate a total analog of set colorings; that is, we study set colorings of the total graph of graphs. Given a graph G = ( V, E ), its total graph T ( G ) is the graph whose vertex set is V ∪ E and in which two vertices are adjacent if and only if their corresponding elements in G are adjacent or incident. First, we establish sharp bounds for the set chromatic number of the total graph of a graph. Furthermore, we study the set colorings of the total graph of different families of graphs.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":"77 1-2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41287211","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
Total Colouring of New Classes of Subcubic graphs 一类新的次三次图的全着色
Theory and Applications of Graphs Pub Date : 2022-07-18 DOI: 10.20429/tag.2022.090207
S. G, Velankanni Anthonymuthu
{"title":"Total Colouring of New Classes of Subcubic graphs","authors":"S. G, Velankanni Anthonymuthu","doi":"10.20429/tag.2022.090207","DOIUrl":"https://doi.org/10.20429/tag.2022.090207","url":null,"abstract":"The total chromatic number of a graph G , denoted χ ′′ ( G ), is the least number of colours needed to colour the vertices and the edges of G such that no incident or adjacent elements (vertices or edges) receive the same colour. The popular Total Colouring Conjecture (TCC) posed by Behzad states that, for every simple graph G , χ ′′ ( G ) ≤ ∆( G ) + 2. In this paper, we prove that the total chromatic number for a family of subcubic graphs called cube connected paths and also for a class of subcubic graphs having the property that the vertices are covered by independent triangles are exactly ∆( G ) + 1. More precisely, these two families of subcubic graphs are shown to be Type 1 graphs.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45145340","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
Geodesic bipancyclicity of the Cartesian product of graphs 图的笛卡尔积的测地线双环性
Theory and Applications of Graphs Pub Date : 2022-07-18 DOI: 10.20429/tag.2022.090206
A. V. Shinde, Y. M. Borse
{"title":"Geodesic bipancyclicity of the Cartesian product of graphs","authors":"A. V. Shinde, Y. M. Borse","doi":"10.20429/tag.2022.090206","DOIUrl":"https://doi.org/10.20429/tag.2022.090206","url":null,"abstract":"A cycle containing a shortest path between two vertices u and v in a graph G is called a ( u, v )-geodesic cycle. A connected graph G is geodesic 2-bipancyclic, if every pair of vertices u, v of it is contained in a ( u, v )-geodesic cycle of length l for each even integer l satisfying 2 d + 2 ≤ l ≤ | V ( G ) | , where d is the distance between u and v. In this paper, we prove that the Cartesian product of two geodesic hamiltonian graphs is a geodesic 2-bipancyclic graph. As a consequence, we show that for n ≥ 2 every n -dimensional torus is a geodesic 2-bipancyclic graph.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41718829","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
On p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs and graph operations 关于无对称弧的无环Hamilton有向图的p竞争图和图运算
Theory and Applications of Graphs Pub Date : 2022-07-18 DOI: 10.20429/tag.2022.090203
Kuniharu Yokomura, M. Tsuchiya
{"title":"On p-competition graphs of loopless Hamiltonian digraphs without symmetric arcs and graph operations","authors":"Kuniharu Yokomura, M. Tsuchiya","doi":"10.20429/tag.2022.090203","DOIUrl":"https://doi.org/10.20429/tag.2022.090203","url":null,"abstract":"","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-07-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46991222","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
Prime labelings on planar grid graphs 平面网格图上的素数标注
Theory and Applications of Graphs Pub Date : 2022-03-24 DOI: 10.20429/tag.2022.090106
S. Curran
{"title":"Prime labelings on planar grid graphs","authors":"S. Curran","doi":"10.20429/tag.2022.090106","DOIUrl":"https://doi.org/10.20429/tag.2022.090106","url":null,"abstract":"A graph G is said to be prime if there is a bijective function f : V ( G ) → { 1 , 2 , . . . , | V ( G ) |} such that f ( u ) and f ( v ) are relatively prime whenever u is adjacent to v . It is known that for any prime p and any integer n such that 1 ≤ n ≤ p , there exists a prime labeling on the p × n planar grid graph P p × P n . We show that P p × P n has a prime labeling for any odd prime p and any integer n such that p < n ≤ p 2 . We discuss how this approach may lead to prime labeling on P p × P n for any odd prime p and any positive integer n .","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48326771","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
Chromatic Polynomials of Signed Book Graphs 有符号书图的色多项式
Theory and Applications of Graphs Pub Date : 2022-03-24 DOI: 10.20429/tag.2022.090104
Deepak Sehrawat, B. Bhattacharjya
{"title":"Chromatic Polynomials of Signed Book Graphs","authors":"Deepak Sehrawat, B. Bhattacharjya","doi":"10.20429/tag.2022.090104","DOIUrl":"https://doi.org/10.20429/tag.2022.090104","url":null,"abstract":"For m ≥ 3 and n ≥ 1, the m -cycle book graph B ( m, n ) consists of n copies of the cycle C m with one common edge. In this paper, we prove that (a) the number of switching non-isomorphic signed B ( m, n ) is n + 1, and (b) the chromatic number of a signed B ( m, n ) is either 2 or 3. We also obtain explicit formulas for the chromatic polynomials and the zero-free chromatic polynomials of switching non-isomorphic signed book graphs.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49606564","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
Characterizing Edge Betweenness-Uniform graphs 边间性的刻画——均匀图
Theory and Applications of Graphs Pub Date : 2022-03-24 DOI: 10.20429/tag.2022.090105
Jana Coroničová Hurajová, T. Madaras, D. Narayan
{"title":"Characterizing Edge Betweenness-Uniform graphs","authors":"Jana Coroničová Hurajová, T. Madaras, D. Narayan","doi":"10.20429/tag.2022.090105","DOIUrl":"https://doi.org/10.20429/tag.2022.090105","url":null,"abstract":"The betweenness centality of an edge e is, summed over all u, v ∈ V ( G ), the ratio of the number of shortest u, v -paths in G containing e to the number of shortest u, v -paths in G . Graphs whose vertices all have the same edge betweenness centrality are called edge betweeness-uniform . It was recently shown by Madaras, Hurajov´a, Newman, Mi-randa, Fl´orez, and Narayan that of the over 11.7 million graphs with ten vertices or fewer, only four graphs are edge betweenness-uniform but not edge-transitive. In this paper we present new results involving properties of betweenness-uniform graphs.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43595867","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
Application of the Combinatorial Nullstellensatz to Integer-magic Graph Labelings 组合nullstellensz在整幻图标记中的应用
Theory and Applications of Graphs Pub Date : 2022-02-22 DOI: 10.20429/tag.2022.090103
R. Low, D. Roberts
{"title":"Application of the Combinatorial Nullstellensatz to Integer-magic Graph Labelings","authors":"R. Low, D. Roberts","doi":"10.20429/tag.2022.090103","DOIUrl":"https://doi.org/10.20429/tag.2022.090103","url":null,"abstract":"Let A be a nontrivial additive abelian group and A∗ = A{0}. A graph is A-magic if there exists an edge labeling f using elements of A∗ which induces a constant vertex labeling of the graph. Here, the induced label on a vertex is obtained by calculating the sum of the edge labels adjacent to that vertex. Such a labeling f is called an A-magic labeling and the constant value of the induced vertex labeling is called an A-magic value. In this paper, we use the Combinatorial Nullstellensatz to show the existence of Zp-magic labelings (prime p ≥ 3 ) for various graphs, without having to construct the Zp-magic labelings. Through many examples, we illustrate the usefulness and limitations in applying the Combinatorial Nullstellensatz to the integer-magic labeling problem. Finally, we focus on Z3-magic labelings and give some results for various classes of graphs.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44209103","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
Facial Achromatic Number of Triangulations with Given Guarding Number 给定保护数的三角形的面消色差数
Theory and Applications of Graphs Pub Date : 2022-02-16 DOI: 10.20429/tag.2022.090101
Naoki Matsumoto, Yumiko Ohno
{"title":"Facial Achromatic Number of Triangulations with Given Guarding Number","authors":"Naoki Matsumoto, Yumiko Ohno","doi":"10.20429/tag.2022.090101","DOIUrl":"https://doi.org/10.20429/tag.2022.090101","url":null,"abstract":"A (not necessarily proper) k -coloring c : V ( G ) ! f 1 ; 2 ; : : : ; k g of a graph G on a surface is a facial t -complete k -coloring if every t -tuple of colors appears on the boundary of some face of G . The maximum number k such that G has a facial t complete k -coloring is called a facial t -achromatic number of G , denoted by t ( G ). In this paper, we investigate the relation between the facial 3-achromatic number and guarding number of triangulations on a surface, where a guarding number of a graph G embedded on a surface, denoted by guard( G ), is the smallest size of its guarding set which is a generalized concept of guards in the art gallery problem. We show that for any graph G embedded on a surface, where ∆( G (cid:3) ) is the largest face size of G . Furthermore, we investigate sufficient conditions for a triangulation G on a surface to satisfy 3 ( G ) = guard( G ) + 2. In particular, we prove that every triangulation G on the sphere with guard( G ) = 2 satis(cid:12)es the above equality and that for one with guarding number 3, it also satis(cid:12)es the above equality with sufficiently large number of vertices.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-02-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48851981","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
Counting Power Domination Sets in Complete m-ary Trees 完全m-玛利树中的计数权力支配集
Theory and Applications of Graphs Pub Date : 2022-01-22 DOI: 10.20429/tag.2023.100104
Sviatlana Kniahnitskaya, Michele Ortiz, Olivia Ramirez, Katharine Shultis, Hays Whitlatch
{"title":"Counting Power Domination Sets in Complete m-ary Trees","authors":"Sviatlana Kniahnitskaya, Michele Ortiz, Olivia Ramirez, Katharine Shultis, Hays Whitlatch","doi":"10.20429/tag.2023.100104","DOIUrl":"https://doi.org/10.20429/tag.2023.100104","url":null,"abstract":"The study of power domination sets arises from the monitoring of electrical network using Phase Measurement Units (PMUs or monitors). This problem was first studied in terms of graphs in [4] in 2002 and has been a topic of much interest since then (see e.g. [1–3, 6, 7]). A PMU placed at a network node measures the voltage at the node and all current phasors at the node [1], and subsequently measures the voltage at some neighboring nodes using the propagation rules described in Definition 1. Since PMUs are expensive, it is desirable to find the minimum number of PMUs needed to monitor a network. This problem is known to be to be NP-complete even for planar bipartite graphs ([3]). Since the cost of technology typically decreases but the cost of employment increases, it is feasible that the cost of placing extra PMUs is preferred to the cost of determining the minimum number of PMUs and an optimal placement. Thus, in this paper, we begin to investigate how probable it is that a randomly placed set of k PMUs will monitor a network.","PeriodicalId":37096,"journal":{"name":"Theory and Applications of Graphs","volume":" ","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45632493","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
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