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A New Upper Bound for the Perfect Italian Domination Number of a Tree 树的完全意大利支配数的一个新上界
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-06-29 DOI: 10.7151/dmgt.2324
S. Nazari-Moghaddam, M. Chellali
{"title":"A New Upper Bound for the Perfect Italian Domination Number of a Tree","authors":"S. Nazari-Moghaddam, M. Chellali","doi":"10.7151/dmgt.2324","DOIUrl":"https://doi.org/10.7151/dmgt.2324","url":null,"abstract":"Abstract A perfect Italian dominating function (PIDF) on a graph G is a function f : V (G) → {0, 1, 2} satisfying the condition that for every vertex u with f(u) = 0, the total weight of f assigned to the neighbors of u is exactly two. The weight of a PIDF is the sum of its functions values over all vertices. The perfect Italian domination number of G, denoted γIp(G) gamma _I^pleft( G right) , is the minimum weight of a PIDF of G. In this paper, we show that for every tree T of order n ≥ 3, with ℓ(T) leaves and s(T) support vertices, γpI(T) ≤ γIp(T)≤4n-l(T)+2s(T-1)5 gamma _I^pleft( T right) le {{4n - mathcal{l}left( T right) + 2sleft( {T - 1} right)} over 5} , improving a previous bound given by T.W. Haynes and M.A. Henning in [Perfect Italian domination in trees, Discrete Appl. Math. 260 (2019) 164–177].","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41726303","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Ascending Subgraph Decompositions of Oriented Graphs that Factor into Triangles 因子为三角形的有向图的上升子图分解
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-06-29 DOI: 10.7151/dmgt.2306
Andrea D. Austin, Brian C. Wagner
{"title":"Ascending Subgraph Decompositions of Oriented Graphs that Factor into Triangles","authors":"Andrea D. Austin, Brian C. Wagner","doi":"10.7151/dmgt.2306","DOIUrl":"https://doi.org/10.7151/dmgt.2306","url":null,"abstract":"Abstract In 1987, Alavi, Boals, Chartrand, Erdős, and Oellermann conjectured that all graphs have an ascending subgraph decomposition (ASD). In a previous paper, Wagner showed that all oriented complete balanced tripartite graphs have an ASD. In this paper, we will show that all orientations of an oriented graph that can be factored into triangles with a large portion of the triangles being transitive have an ASD. We will also use the result to obtain an ASD for any orientation of complete multipartite graphs with 3n partite classes each containing 2 vertices (a K(2 : 3n)) or 4 vertices (a K(4 : 3n)).","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43103878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Bounds on Domination Parameters in Graphs: A Brief Survey 图中支配参数的界限:简要综述
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-06-29 DOI: 10.7151/dmgt.2454
Michael A. Henning
{"title":"Bounds on Domination Parameters in Graphs: A Brief Survey","authors":"Michael A. Henning","doi":"10.7151/dmgt.2454","DOIUrl":"https://doi.org/10.7151/dmgt.2454","url":null,"abstract":"Abstract In this paper we present a brief survey of bounds on selected domination parameters. We focus primarily on bounds on domination parameters in terms of the order and minimum degree of the graph. We present a list of open problems and conjectures that have yet to be solved in the hope of attracting future researchers to the field.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43388752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Oriented Chromatic Number of Cartesian Products Pm □ Pn and Cm □ Pn 笛卡尔积的取向色数Pm□Pn和Cm□Pn
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-06-29 DOI: 10.7151/dmgt.2307
Anna Nenca
{"title":"Oriented Chromatic Number of Cartesian Products Pm □ Pn and Cm □ Pn","authors":"Anna Nenca","doi":"10.7151/dmgt.2307","DOIUrl":"https://doi.org/10.7151/dmgt.2307","url":null,"abstract":"Abstract We consider oriented chromatic number of Cartesian products of two paths Pm □ Pn and of Cartesian products of paths and cycles, Cm □ Pn. We say that the oriented graph G→ vec G is colored by an oriented graph H→ vec H if there is a homomorphism from G→ vec G to H→ vec H . In this paper we show that there exists an oriented tournament H→10 {vec H_{10}} with ten vertices which colors every orientation of P8 □ Pn and every orientation of Cm □ Pn, for m = 3, 4, 5, 6, 7 and n ≥ 1. We also show that there exists an oriented graph T→16 {vec T_{16}} with sixteen vertices which colors every orientation of Cm □ Pn.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47822802","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Extremal Digraphs Avoiding Distinct Walks of Length 4 with the Same Endpoints 极值有向图避免具有相同端点的长度为4的不同行走
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-06-29 DOI: 10.7151/dmgt.2321
Zhenhua Lyu
{"title":"Extremal Digraphs Avoiding Distinct Walks of Length 4 with the Same Endpoints","authors":"Zhenhua Lyu","doi":"10.7151/dmgt.2321","DOIUrl":"https://doi.org/10.7151/dmgt.2321","url":null,"abstract":"Abstract Let n ≥ 8 be an integer. We characterize the extremal digraphs of order n with the maximum number of arcs avoiding distinct walks of length 4 with the same endpoints.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45447183","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 8
On the k-Independence Number of Graph Products 论图积的k独立数
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-05-31 DOI: 10.7151/dmgt.2480
A. Abiad, Hidde Koerts
{"title":"On the k-Independence Number of Graph Products","authors":"A. Abiad, Hidde Koerts","doi":"10.7151/dmgt.2480","DOIUrl":"https://doi.org/10.7151/dmgt.2480","url":null,"abstract":"Abstract The k-independence number of a graph, αk(G), is the maximum size of a set of vertices at pairwise distance greater than k, or alternatively, the independence number of the k-th power graph Gk. Although it is known that αk(G) = α(Gk), this, in general, does not hold for most graph products, and thus the existing bounds for α of graph products cannot be used. In this paper we present sharp upper bounds for the k-independence number of several graph products. In particular, we focus on the Cartesian, tensor, strong, and lexicographic products. Some of the bounds previously known in the literature for k = 1 follow as corollaries of our main results.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-05-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42677082","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
The Linear Arboricity of Graphs with Low Treewidth 低树宽图的线性树性
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-05-27 DOI: 10.7151/dmgt.2456
Xiang Tan, Jian-Liang Wu
{"title":"The Linear Arboricity of Graphs with Low Treewidth","authors":"Xiang Tan, Jian-Liang Wu","doi":"10.7151/dmgt.2456","DOIUrl":"https://doi.org/10.7151/dmgt.2456","url":null,"abstract":"Abstract Let G be a graph with treewidth k. In the paper, it is proved that if k ≤ 3 and maximum degree Δ ≥ 5, or k = 4 and Δ ≥ 9, or Δ ≥ 4k − 3 and k ≥ 5, then the linear arboricity la(G) of G is ⌈ Δ2 ⌉ leftlceil {{Delta over 2}} rightrceil","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48001799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Degree Sum Condition for Vertex-Disjoint 5-Cycles 顶点不连续5-环的度和条件
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-05-27 DOI: 10.7151/dmgt.2458
Maoqun Wang, Jianguo Qian
{"title":"Degree Sum Condition for Vertex-Disjoint 5-Cycles","authors":"Maoqun Wang, Jianguo Qian","doi":"10.7151/dmgt.2458","DOIUrl":"https://doi.org/10.7151/dmgt.2458","url":null,"abstract":"Abstract Let n and k be two integers and G a graph with n = 5k vertices. Wang proved that if δ (G) ≥ 3k, then G contains k vertex disjoint cycles of length 5. In 2018, Chiba and Yamashita asked whether the degree condition can be replaced by degree sum condition. In this paper, we give a positive answer to this question.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48138607","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
L(2, 1)-Labeling of the Iterated Mycielski Graphs of Graphs and Some Problems Related to Matching Problems 图的迭代Mycielski图的L（2，1）-标记及与匹配问题有关的一些问题
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-05-27 DOI: 10.7151/dmgt.2457
Kamal Dliou, H. El Boujaoui, M. Kchikech
{"title":"L(2, 1)-Labeling of the Iterated Mycielski Graphs of Graphs and Some Problems Related to Matching Problems","authors":"Kamal Dliou, H. El Boujaoui, M. Kchikech","doi":"10.7151/dmgt.2457","DOIUrl":"https://doi.org/10.7151/dmgt.2457","url":null,"abstract":"Abstract In this paper, we study the L(2, 1)-labeling of the Mycielski graph and the iterated Mycielski graph of graphs in general. For a graph G and all t ≥ 1, we give sharp bounds for λ(Mt(G)) the L(2, 1)-labeling number of the t-th iterated Mycielski graph in terms of the number of iterations t, the order n of G, the maximum degree Δ, and λ(G) the L(2, 1)-labeling number of G. For t = 1, we present necessary and sufficient conditions between the 4-star matching number of the complement graph and λ(M(G)) the L(2, 1)-labeling number of the Mycielski graph of a graph, with some applications to special graphs. For all t ≥ 2, we prove that for any graph G of order n, we have 2t−1(n + 2) − 2 ≤ λ(Mt(G)) ≤ 2t(n + 1) − 2. Thereafter, we characterize the graphs achieving the upper bound 2t(n+1)−2, then by using the Marriage Theorem and Tutte’s characterization of graphs with a perfect 2-matching, we characterize all graphs without isolated vertices achieving the lower bound 2t−1(n + 2) − 2. We determine the L(2, 1)-labeling number for the Mycielski graph and the iterated Mycielski graph of some graph classes.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44261855","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Linear Arboricity of 1-Planar Graphs 1-平面图的线性拟合性
IF 0.7 4区 数学
Discussiones Mathematicae Graph Theory Pub Date : 2022-05-27 DOI: 10.7151/dmgt.2453
Weifan Wang, Juan Liu, Yiqiao Wang
{"title":"Linear Arboricity of 1-Planar Graphs","authors":"Weifan Wang, Juan Liu, Yiqiao Wang","doi":"10.7151/dmgt.2453","DOIUrl":"https://doi.org/10.7151/dmgt.2453","url":null,"abstract":"Abstract The linear arboricity la(G) of a graph G is the minimum number of linear forests that partition the edges of G. In 1981, Akiyama, Exoo and Harary conjectured that ⌈ Δ(G)2 ⌉≤la(G)≤⌈ Δ(G)+12 ⌉ leftlceil {{{Delta left( G right)} over 2}} rightrceil le laleft( G right) le leftlceil {{{Delta left( G right) + 1} over 2}} rightrceil for any simple graph G. A graph G is 1-planar if it can be drawn in the plane so that each edge has at most one crossing. In this paper, we confirm the conjecture for 1-planar graphs G with Δ(G) ≥ 13.","PeriodicalId":48875,"journal":{"name":"Discussiones Mathematicae Graph Theory","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2022-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46861447","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","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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