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].
图G上的一个完全意大利支配函数(PIDF)是一个函数f: V (G)→{0,1,2}，满足对于f(u) = 0的每个顶点u，分配给u的邻点f的总权值恰好为2。PIDF的权重是其所有顶点上的函数值的和。G的完美义大利支配数，记作γIp(G) gamma ＿I^p left (G right)，是G的PIDF的最小权值。在本文中，我们证明了对于n≥3阶的树T，有n (T)个叶子和s(T)个支持顶点，γpI(T)≤γIp(T)≤4n-l(T)+2s(T-1)5 gamma ＿I^p left (T right) le 4n-{{mathcal{l}left (T right)+2s left ({T-1}right) }over 5，}改进了T.W. Haynes和M.A. Henning在[树的完美意大利支配，离散应用]中给出的前界。数学，260(2019)164-177]。
{"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}
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)).
{"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}
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.
{"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}
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.
{"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}
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.
设n≥8为整数。我们用最大数量的弧来刻画n阶的极值有向图，以避免相同端点的长度为4的不同行走。
{"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}
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.
{"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}
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
{"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}
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.
{"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}
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.
{"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}
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.
图G的线性树性la(G)是划分G边的线性森林的最小个数。1981年，Akiyama, Exoo和Harary推测出了≤Δ(G)2≤la(G)≤≤≤Δ(G)+12 leftlceil {{{Delta left(g) right）} over 2}} rightrceil le 拉left(g) right） le leftlceil {{{Delta left(g) right) + 1} over 2}} rightrceil 对于任何简单图G，如果图G可以在平面上画出来，使得每条边最多有一个交叉点，那么它就是一个平面图G。本文证实了对于Δ(G)≥13的1-平面图G的猜想。
{"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}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：