Pub Date : 2024-10-01DOI: 10.1016/j.dam.2024.09.024
Yuanrui Feng , Jun Ge , Douglas B. West , Yan Yang
Visibility representation of digraphs was introduced by Axenovich et al. (2013) as a natural generalization of -bar visibility representation of undirected graphs. A -bar visibility representation of a digraph assigns each vertex at most horizontal bars in the plane so that there is an arc in the digraph if and only if some bar for “sees” some bar for above it along an unblocked vertical strip with positive width. The visibility number is the least such that has a -bar visibility representation. In this paper, we solve several problems about posed by Axenovich et al. and prove that determining whether the bar visibility number of a digraph is 2 is NP-complete.
数图的可见性表示是由 Axenovich 等人（2013 年）提出的，是对无向图的 t 条可见性表示的自然概括。数图 G 的 t 条可见度表示法为每个顶点在平面上分配了最多 t 条水平条，因此当且仅当 x 的某个条沿着宽度为正的无阻挡垂直条 "看到 "其上方 y 的某个条时，数图中才有弧 xy。可见度数 b(G) 是 G 具有 t 条可见度表示的最小 t。在本文中，我们解决了阿克森诺维奇等人提出的几个关于 b(G) 的问题，并证明确定一个数图的条形可见度数是否为 2 是 NP-完全的。
{"title":"Some new results on bar visibility of digraphs","authors":"Yuanrui Feng , Jun Ge , Douglas B. West , Yan Yang","doi":"10.1016/j.dam.2024.09.024","DOIUrl":"10.1016/j.dam.2024.09.024","url":null,"abstract":"<div><div>Visibility representation of digraphs was introduced by Axenovich et al. (2013) as a natural generalization of <span><math><mi>t</mi></math></span>-bar visibility representation of undirected graphs. A <span><math><mi>t</mi></math></span><em>-bar visibility representation</em> of a digraph <span><math><mi>G</mi></math></span> assigns each vertex at most <span><math><mi>t</mi></math></span> horizontal bars in the plane so that there is an arc <span><math><mrow><mi>x</mi><mi>y</mi></mrow></math></span> in the digraph if and only if some bar for <span><math><mi>x</mi></math></span> “sees” some bar for <span><math><mi>y</mi></math></span> above it along an unblocked vertical strip with positive width. The <em>visibility number</em> <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the least <span><math><mi>t</mi></math></span> such that <span><math><mi>G</mi></math></span> has a <span><math><mi>t</mi></math></span>-bar visibility representation. In this paper, we solve several problems about <span><math><mrow><mi>b</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> posed by Axenovich et al. and prove that determining whether the bar visibility number of a digraph is 2 is NP-complete.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 342-352"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424795","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-10-01DOI: 10.1016/j.dam.2024.09.025
Qingqiong Cai , Shinya Fujita , Henry Liu , Boram Park
An edge-coloured path is monochromatic if all of its edges have the same colour. For a -connected graph , the monochromatic-connection number of , denoted by , is the maximum number of colours in an edge-colouring of such that, any two vertices are connected by internally vertex-disjoint monochromatic paths. In this paper, we shall study the parameter . We obtain bounds for , for general graphs . We also compute exactly when is small, and is a graph on vertices, with a spanning -connected subgraph having the minimum possible number of edges, namely . We prove a similar result when is a bipartite graph.
如果一条边缘着色的路径的所有边缘颜色相同，那么这条路径就是单色的。对于 k 个连接图 G，G 的单色 k 连接数（用 mck(G) 表示）是指在 G 的边缘颜色中，任意两个顶点通过 k 个内部顶点相交的单色路径连接的最大颜色数。本文将研究 mck(G) 参数。对于一般图 G，我们得到了 mck(G) 的边界。当 k 很小时，我们也能精确计算 mck(G)。当 G 是 n 个顶点上的图时，有一个跨 k 连接的子图具有尽可能少的边数，即 ⌈kn2⌉。当 G 是双向图时，我们会证明类似的结果。
{"title":"Monochromatic k-connection of graphs","authors":"Qingqiong Cai , Shinya Fujita , Henry Liu , Boram Park","doi":"10.1016/j.dam.2024.09.025","DOIUrl":"10.1016/j.dam.2024.09.025","url":null,"abstract":"<div><div>An edge-coloured path is <em>monochromatic</em> if all of its edges have the same colour. For a <span><math><mi>k</mi></math></span>-connected graph <span><math><mi>G</mi></math></span>, the <em>monochromatic</em> <span><math><mi>k</mi></math></span><em>-connection number</em> of <span><math><mi>G</mi></math></span>, denoted by <span><math><mrow><mi>m</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is the maximum number of colours in an edge-colouring of <span><math><mi>G</mi></math></span> such that, any two vertices are connected by <span><math><mi>k</mi></math></span> internally vertex-disjoint monochromatic paths. In this paper, we shall study the parameter <span><math><mrow><mi>m</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We obtain bounds for <span><math><mrow><mi>m</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, for general graphs <span><math><mi>G</mi></math></span>. We also compute <span><math><mrow><mi>m</mi><msub><mrow><mi>c</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> exactly when <span><math><mi>k</mi></math></span> is small, and <span><math><mi>G</mi></math></span> is a graph on <span><math><mi>n</mi></math></span> vertices, with a spanning <span><math><mi>k</mi></math></span>-connected subgraph having the minimum possible number of edges, namely <span><math><mrow><mo>⌈</mo><mfrac><mrow><mi>k</mi><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow></math></span>. We prove a similar result when <span><math><mi>G</mi></math></span> is a bipartite graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 328-341"},"PeriodicalIF":1.0,"publicationDate":"2024-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142424794","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-30DOI: 10.1016/j.dam.2024.09.023
Stasys Jukna
A monotone Boolean circuit computing a monotone Boolean function is a read- circuit if the polynomial produced (purely syntactically) by the arithmetic version of the circuit has the property that for every prime implicant of , the polynomial contains at least one monomial with the same set of variables, each appearing with degree . Every monotone circuit is a read- circuit for some . We show that already read-1 circuits are not weaker than monotone arithmetic constant-free circuits computing multilinear polynomials, are not weaker than non-monotone multilinear circuits computing monotone Boolean functions, and have the same power as tropical circuits solving minimization problems. Finally, we show that read-2 circuits can be exponentially smaller than read-1 circuits.
{"title":"Notes on Boolean read-k and multilinear circuits","authors":"Stasys Jukna","doi":"10.1016/j.dam.2024.09.023","DOIUrl":"10.1016/j.dam.2024.09.023","url":null,"abstract":"<div><div>A monotone Boolean <span><math><mrow><mo>(</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>)</mo></mrow></math></span> circuit computing a monotone Boolean function <span><math><mi>f</mi></math></span> is a read-<span><math><mi>k</mi></math></span> circuit if the polynomial produced (purely syntactically) by the arithmetic <span><math><mrow><mo>(</mo><mo>+</mo><mo>,</mo><mo>×</mo><mo>)</mo></mrow></math></span> version of the circuit has the property that for every prime implicant of <span><math><mi>f</mi></math></span>, the polynomial contains at least one monomial with the same set of variables, each appearing with degree <span><math><mrow><mo>⩽</mo><mi>k</mi></mrow></math></span>. Every monotone circuit is a read-<span><math><mi>k</mi></math></span> circuit for some <span><math><mi>k</mi></math></span>. We show that already read-1 <span><math><mrow><mo>(</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>)</mo></mrow></math></span> circuits are not weaker than monotone arithmetic constant-free <span><math><mrow><mo>(</mo><mo>+</mo><mo>,</mo><mo>×</mo><mo>)</mo></mrow></math></span> circuits computing multilinear polynomials, are not weaker than non-monotone multilinear <span><math><mrow><mo>(</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>,</mo><mo>¬</mo><mo>)</mo></mrow></math></span> circuits computing monotone Boolean functions, and have the same power as tropical <span><math><mrow><mo>(</mo><mo>min</mo><mo>,</mo><mo>+</mo><mo>)</mo></mrow></math></span> circuits solving <span><math><mrow><mn>0</mn><mo>/</mo><mn>1</mn></mrow></math></span> minimization problems. Finally, we show that read-2 <span><math><mrow><mo>(</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>)</mo></mrow></math></span> circuits can be exponentially smaller than read-1 <span><math><mrow><mo>(</mo><mo>∨</mo><mo>,</mo><mo>∧</mo><mo>)</mo></mrow></math></span> circuits.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 307-327"},"PeriodicalIF":1.0,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142357013","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-30DOI: 10.1016/j.dam.2024.09.018
Yasemin Büyükçolak , Didem Gözüpek , Sibel Özkan
A graph is called well-indumatched if all of its maximal induced matchings have the same size. Akbari et al. (0000) provided a characterization of well-indumatched trees and some results on well-indumatched unicyclic graphs. In this paper, we extend this result by providing a complete structural characterization of well-indumatched pseudotrees, in which each component has no cycle or a unique cycle, by identifying the well-indumatched graph families.
{"title":"Well-indumatched Pseudoforests","authors":"Yasemin Büyükçolak , Didem Gözüpek , Sibel Özkan","doi":"10.1016/j.dam.2024.09.018","DOIUrl":"10.1016/j.dam.2024.09.018","url":null,"abstract":"<div><div>A graph is called well-indumatched if all of its maximal induced matchings have the same size. Akbari et al. (0000) provided a characterization of well-indumatched trees and some results on well-indumatched unicyclic graphs. In this paper, we extend this result by providing a complete structural characterization of well-indumatched pseudotrees, in which each component has no cycle or a unique cycle, by identifying the well-indumatched graph families.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 85-102"},"PeriodicalIF":1.0,"publicationDate":"2024-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142359429","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-28DOI: 10.1016/j.dam.2024.09.016
Weimin Qian, Feng Li
Effectiveness represents a fundamental performance metric in network communication systems and is intricately tied to the routing made within the network. Since the vertex forwarding index is primarily employed to measure the load-bearing capacity of network nodes, it has become a crucial parameter for evaluating the quality of network routing. By investigating the topological parameters and properties of the Strong product network and its factor networks (The undirected cycle networks with vertices and the complete networks with vertices), we have determined the exact vertex forwarding index of Strong product graph , which depends solely on the number of vertices in the factor networks.
{"title":"Exact vertex forwarding index of the Strong product of complete graph and cycle","authors":"Weimin Qian, Feng Li","doi":"10.1016/j.dam.2024.09.016","DOIUrl":"10.1016/j.dam.2024.09.016","url":null,"abstract":"<div><div>Effectiveness represents a fundamental performance metric in network communication systems and is intricately tied to the routing made within the network. Since the vertex forwarding index is primarily employed to measure the load-bearing capacity of network nodes, it has become a crucial parameter for evaluating the quality of network routing. By investigating the topological parameters and properties of the Strong product network <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⊗</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span> and its factor networks (The undirected cycle networks <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>c</mi></mrow></msub></math></span> with <span><math><mrow><mi>c</mi><mo>≥</mo><mn>3</mn></mrow></math></span> vertices and the complete networks <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span> vertices), we have determined the exact vertex forwarding index of Strong product graph <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>⊗</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>c</mi></mrow></msub></mrow></math></span>, which depends solely on the number of vertices in the factor networks.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 69-84"},"PeriodicalIF":1.0,"publicationDate":"2024-09-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142358808","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-26DOI: 10.1016/j.dam.2024.09.022
Xiaoyan Lai, Yishuo Shi
Nowadays, maximizing the non-negative and non-submodular objective functions under Knapsack constraint or Cardinality constraint is deeply researched. Nevertheless, few studies study the objective functions with non-submodularity under the non-submodular constraint. And there are many practical applications of the situations, such as Epidemic transmission, and Sensor Placement and Feature Selection problem. In this paper, we study the maximization of the non-submodular objective functions under the non-submodular constraint. Based on the non-submodular constraint, we discuss the maximization of the objective functions with some specific properties, which includes the property of negative, and then, we obtain the corresponding approximate ratios by the greedy algorithm. What is more, these approximate ratios could be improved when the constraint becomes tight.
{"title":"Approximation algorithm of maximizing non-submodular functions under non-submodular constraint","authors":"Xiaoyan Lai, Yishuo Shi","doi":"10.1016/j.dam.2024.09.022","DOIUrl":"10.1016/j.dam.2024.09.022","url":null,"abstract":"<div><div>Nowadays, maximizing the non-negative and non-submodular objective functions under Knapsack constraint or Cardinality constraint is deeply researched. Nevertheless, few studies study the objective functions with non-submodularity under the non-submodular constraint. And there are many practical applications of the situations, such as Epidemic transmission, and Sensor Placement and Feature Selection problem. In this paper, we study the maximization of the non-submodular objective functions under the non-submodular constraint. Based on the non-submodular constraint, we discuss the maximization of the objective functions with some specific properties, which includes the property of negative, and then, we obtain the corresponding approximate ratios by the greedy algorithm. What is more, these approximate ratios could be improved when the constraint becomes tight.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 48-68"},"PeriodicalIF":1.0,"publicationDate":"2024-09-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142324153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-23DOI: 10.1016/j.dam.2024.09.013
Shyama S. , Radha R. Iyer
The honeycomb mesh, based on the hexagonal tessellation, is considered a multiprocessor interconnection network. Cellular phone station placement, computer graphics and image processing are some applications where hexagonal tessellations are used. They are also used for the representation of benzenoid hydrocarbons. In this paper, we study the irregular chromatic number of Honeycomb Network, Honeycomb Cup Network and Honeycomb Torus and establish an exact value for the same. Also, we find an upper bound for the Honeycomb Rhombic and Rectangular Torus networks.
{"title":"Unraveling the enigmatic irregular coloring of Honeycomb Networks","authors":"Shyama S. , Radha R. Iyer","doi":"10.1016/j.dam.2024.09.013","DOIUrl":"10.1016/j.dam.2024.09.013","url":null,"abstract":"<div><div>The honeycomb mesh, based on the hexagonal tessellation, is considered a multiprocessor interconnection network. Cellular phone station placement, computer graphics and image processing are some applications where hexagonal tessellations are used. They are also used for the representation of benzenoid hydrocarbons. In this paper, we study the irregular chromatic number of Honeycomb Network, Honeycomb Cup Network and Honeycomb Torus and establish an exact value for the same. Also, we find an upper bound for the Honeycomb Rhombic and Rectangular Torus networks.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"360 ","pages":"Pages 282-296"},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312969","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-23DOI: 10.1016/j.dam.2024.09.014
Jing Gao, Xueliang Li, Ning Yang, Ruiling Zheng
The energy of a graph is the sum of the absolute values of all eigenvalues of its adjacency matrix. Let be the graph obtained from two copies of joined by a path . In 2001, Gutman and Vidović (2001) conjectured that the bicyclic graph with the maximal energy is . This conjecture is true for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, Ji and Li (2012) proved the conjecture for bicyclic graphs which have exactly two edge-disjoint cycles such that one of them is even and the other is odd. This paper is to prove the conjecture for bicyclic graphs containing two odd cycles with one common vertex.
{"title":"Maximum energy bicyclic graphs containing two odd cycles with one common vertex","authors":"Jing Gao, Xueliang Li, Ning Yang, Ruiling Zheng","doi":"10.1016/j.dam.2024.09.014","DOIUrl":"10.1016/j.dam.2024.09.014","url":null,"abstract":"<div><div>The energy of a graph is the sum of the absolute values of all eigenvalues of its adjacency matrix. Let <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>6</mn><mo>,</mo><mn>6</mn></mrow></msubsup></math></span> be the graph obtained from two copies of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub></math></span> joined by a path <span><math><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>10</mn></mrow></msub></math></span>. In 2001, Gutman and Vidović (2001) conjectured that the bicyclic graph with the maximal energy is <span><math><msubsup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow><mrow><mn>6</mn><mo>,</mo><mn>6</mn></mrow></msubsup></math></span>. This conjecture is true for bipartite bicyclic graphs. For non-bipartite bicyclic graphs, Ji and Li (2012) proved the conjecture for bicyclic graphs which have exactly two edge-disjoint cycles such that one of them is even and the other is odd. This paper is to prove the conjecture for bicyclic graphs containing two odd cycles with one common vertex.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 1-21"},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-23DOI: 10.1016/j.dam.2024.09.015
Kamal Dliou
<div><div>Let <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote respectively the independence, matching, 2-matching and packing chromatic numbers of a graph <span><math><mi>G</mi></math></span>. A well-known construction on graphs, called the Mycielskian of a graph, transforms any <span><math><mi>k</mi></math></span>-chromatic graph <span><math><mi>G</mi></math></span> into a <span><math><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-chromatic graph <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> having an equal clique number to <span><math><mi>G</mi></math></span>. The <span><math><mi>t</mi></math></span>th iterated Mycielskian of a graph <span><math><mi>G</mi></math></span>, denoted <span><math><mrow><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is obtained by iteratively repeating the Mycielskian transformation <span><math><mi>t</mi></math></span> times. In this paper, we give <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> in terms of <span><math><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Then we show that for all <span><math><mrow><mi>t</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mo>max</mo><mrow><mo>{</mo><mrow><mo>|</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>α</mi><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>. We characterize for all <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, the connected graphs having <span><math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> and those having <span><math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>
{"title":"Independence, matching and packing coloring of the iterated Mycielskian of graphs","authors":"Kamal Dliou","doi":"10.1016/j.dam.2024.09.015","DOIUrl":"10.1016/j.dam.2024.09.015","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><mi>ν</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>ρ</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> denote respectively the independence, matching, 2-matching and packing chromatic numbers of a graph <span><math><mi>G</mi></math></span>. A well-known construction on graphs, called the Mycielskian of a graph, transforms any <span><math><mi>k</mi></math></span>-chromatic graph <span><math><mi>G</mi></math></span> into a <span><math><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></math></span>-chromatic graph <span><math><mrow><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> having an equal clique number to <span><math><mi>G</mi></math></span>. The <span><math><mi>t</mi></math></span>th iterated Mycielskian of a graph <span><math><mi>G</mi></math></span>, denoted <span><math><mrow><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, is obtained by iteratively repeating the Mycielskian transformation <span><math><mi>t</mi></math></span> times. In this paper, we give <span><math><mrow><mi>α</mi><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></math></span> in terms of <span><math><mrow><msub><mrow><mi>ν</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Then we show that for all <span><math><mrow><mi>t</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, <span><math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mo>max</mo><mrow><mo>{</mo><mrow><mo>|</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>α</mi><mrow><mo>(</mo><mi>M</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>. We characterize for all <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, the connected graphs having <span><math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><msup><mrow><mi>M</mi></mrow><mrow><mi>t</mi><mo>−</mo><mn>1</mn></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span> and those having <span><math><mrow><mi>α</mi><mrow><mo>(</mo><msup><mrow><mi>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 22-33"},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312555","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-09-23DOI: 10.1016/j.dam.2024.09.019
Jing Lou , Ruifang Liu , Jinlong Shu
The toughness for The concept of toughness initially proposed by Chvátal in 1973, which serves as a simple way to measure how tightly various pieces of a graph hold together. A graph is called -tough if It is very interesting to investigate the relations between toughness and eigenvalues of graphs. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] provided sufficient conditions in terms of the spectral radius for a graph to be 1-tough with minimum degree and -tough with being an integer, respectively. By using some typical distance spectral techniques and structural analysis, we in this paper present a sufficient condition based on the distance spectral radius to guarantee a graph to be 1-tough with minimum degree Moreover, we also prove sufficient conditions with respect to the distance spectral radius for a graph to be -tough, where or is a positive integer.
G≇Kn 的韧性 τ(G)=min{|S|c(G-S):Sis a cut set of vertices inG} 。韧性的概念最初是由 Chvátal 于 1973 年提出的，它是一种简单的方法来衡量图中各个部分的紧密程度。如果 τ(G)≥t ，则图 G 称为 t-韧性图。研究图的韧性和特征值之间的关系非常有趣。Fan、Lin 和 Lu [European J. Combin.通过使用一些典型的距离谱技术和结构分析，我们在本文中提出了一个基于距离谱半径的充分条件，以保证图是最小度为 δ 的 1-韧图。此外，我们还证明了关于距离谱半径的充分条件，以保证图是 t-韧图，其中 t 或 1t 是正整数。
{"title":"Toughness and distance spectral radius in graphs involving minimum degree","authors":"Jing Lou , Ruifang Liu , Jinlong Shu","doi":"10.1016/j.dam.2024.09.019","DOIUrl":"10.1016/j.dam.2024.09.019","url":null,"abstract":"<div><div>The <em>toughness</em> <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>min</mi><mrow><mo>{</mo><mfrac><mrow><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow></mrow><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></mrow></mfrac><mo>:</mo><mi>S</mi><mspace></mspace><mtext>is a cut set of vertices in</mtext><mspace></mspace><mi>G</mi><mo>}</mo></mrow></mrow></math></span> for <span><math><mrow><mi>G</mi><mo>≇</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>.</mo></mrow></math></span> The concept of toughness initially proposed by Chvátal in 1973, which serves as a simple way to measure how tightly various pieces of a graph hold together. A graph <span><math><mi>G</mi></math></span> is called <span><math><mi>t</mi></math></span><em>-tough</em> if <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mi>t</mi><mo>.</mo></mrow></math></span> It is very interesting to investigate the relations between toughness and eigenvalues of graphs. Fan, Lin and Lu [European J. Combin. 110 (2023) 103701] provided sufficient conditions in terms of the spectral radius for a graph to be 1-tough with minimum degree <span><math><mi>δ</mi></math></span> and <span><math><mi>t</mi></math></span>-tough with <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span> being an integer, respectively. By using some typical distance spectral techniques and structural analysis, we in this paper present a sufficient condition based on the distance spectral radius to guarantee a graph to be 1-tough with minimum degree <span><math><mrow><mi>δ</mi><mo>.</mo></mrow></math></span> Moreover, we also prove sufficient conditions with respect to the distance spectral radius for a graph to be <span><math><mi>t</mi></math></span>-tough, where <span><math><mi>t</mi></math></span> or <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mi>t</mi></mrow></mfrac></math></span> is a positive integer.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 34-47"},"PeriodicalIF":1.0,"publicationDate":"2024-09-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142312390","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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