This is a collection of open problems related to/presented at the 1st Chinese–Southeasteuropean conference on discrete mathematics and applications that was held at Serbian Academy of Sciences and Arts in Belgrade, Serbia, from June 9–14, 2024.
{"title":"Research problems from the 1st Chinese–Southeasteuropean conference on discrete mathematics and applications","authors":"Vedran Krčadinac , Shenggui Zhang , Liming Xiong , Dragan Stevanović","doi":"10.1016/j.dam.2025.02.037","DOIUrl":"10.1016/j.dam.2025.02.037","url":null,"abstract":"<div><div>This is a collection of open problems related to/presented at the 1st Chinese–Southeasteuropean conference on discrete mathematics and applications that was held at Serbian Academy of Sciences and Arts in Belgrade, Serbia, from June 9–14, 2024.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 99-104"},"PeriodicalIF":1.0,"publicationDate":"2025-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143739099","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 : 2025-03-30DOI: 10.1016/j.dam.2025.03.019
Mingzu Zhang , Hongxi Liu , Chia-Wei Lee , Weihua Yang
<div><div>The edge isopermetric problem on hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, proposed by Harper in 1964, is to find a vertex subset with cardinality <span><math><mi>m</mi></math></span> in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, such that the edge cut separating any vertex subset with cardinality <span><math><mi>m</mi></math></span> from its complement has minimum size. Since Harper, Lindsey, Bernstein and Hart solved the edge isoperimetric problem of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> by lexicographic order, the edge isoperimetric problem is intimately tied to many-to-many disjoint paths problem. The maximum cardinality of edge disjoint paths connecting any two disjoint connected subgraphs of order <span><math><mi>h</mi></math></span> in a connected graph <span><math><mi>G</mi></math></span> can be defined by the minimum modified edge-cut, called the <span><math><mi>h</mi></math></span>-extra edge-connectivity of <span><math><mi>G</mi></math></span>. It is the cardinality of the minimum set of edges in a connected graph <span><math><mi>G</mi></math></span>, if such a set exists, whose deletion disconnects <span><math><mi>G</mi></math></span> and leaves every remaining component with at least <span><math><mi>h</mi></math></span> vertices. The <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-enhanced hypercubes <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> are constructed by adding a matching between some pair copies of <span><math><mi>k</mi></math></span> dimensional subcubes <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <span><math><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>. The distribution of the values of the <span><math><mi>h</mi></math></span>-extra edge-connectivity on a recursive graph is uneven and presents a concentration phenomenon. In this paper, we start with analysing the fractal properties of the optimal solution of the edge isoperimetric problem of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span>. And it is shown that although the members of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> are not isomorphic to each other according to different <span><math><mi>k</mi></math></span> where <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>, when <span><math><mi>n</mi></math></span> approaches infinity, the <span><math><mi>h</mi></math></span>-extra edge-connectivity of <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-enhanced hypercubes presents a concentration phenome
哈珀于 1964 年提出了超立方体 Qn 上的边等周问题，即在 Qn 中找到一个心数为 m 的顶点子集，使得任何心数为 m 的顶点子集与其补集之间的边切最小。由于哈珀、林赛、伯恩斯坦和哈特按词典顺序解决了 Qn 的边等周问题，因此边等周问题与多对多不相交路径问题密切相关。连接连通图 G 中任何两个阶数为 h 的互不相邻子图的边互不相交路径的最大卡片数，可以用最小修正边切定义，称为 G 的 h-额外边连接性。它是连通图 G 中最小边集的卡片数（如果存在这样的边集），删除该边集后，G 将断开连接，剩下的每个部分至少有 h 个顶点。(n,k)增强超立方体 Qn,k 是通过在 k 维子立方体 Qk 的某些对副本之间添加 1≤k≤n-1 的匹配而构建的。递归图上的 h 外边连通性值的分布是不均匀的，并呈现集中现象。本文首先分析了 Qn,k 边等周问题最优解的分形特性。结果表明，虽然 Qn,k 的成员之间根据 2≤k≤n-1 的不同 k 并不同构，但当 n 接近无穷大时，(n,k)增强超立方体的 h 外边连接性呈现集中现象。也就是说，对于至少 2/3 的 h≤2n-1，相应的精确值 λh(Qn,k)集中在 2n-1/3≤h≤2n-1 的 2n-1 上。
{"title":"Edge isoperimetric method: At least 2/3 of h-extra edge-connectivity of a kind of cube-based graphs concentrates on 2n−1","authors":"Mingzu Zhang , Hongxi Liu , Chia-Wei Lee , Weihua Yang","doi":"10.1016/j.dam.2025.03.019","DOIUrl":"10.1016/j.dam.2025.03.019","url":null,"abstract":"<div><div>The edge isopermetric problem on hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, proposed by Harper in 1964, is to find a vertex subset with cardinality <span><math><mi>m</mi></math></span> in <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, such that the edge cut separating any vertex subset with cardinality <span><math><mi>m</mi></math></span> from its complement has minimum size. Since Harper, Lindsey, Bernstein and Hart solved the edge isoperimetric problem of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> by lexicographic order, the edge isoperimetric problem is intimately tied to many-to-many disjoint paths problem. The maximum cardinality of edge disjoint paths connecting any two disjoint connected subgraphs of order <span><math><mi>h</mi></math></span> in a connected graph <span><math><mi>G</mi></math></span> can be defined by the minimum modified edge-cut, called the <span><math><mi>h</mi></math></span>-extra edge-connectivity of <span><math><mi>G</mi></math></span>. It is the cardinality of the minimum set of edges in a connected graph <span><math><mi>G</mi></math></span>, if such a set exists, whose deletion disconnects <span><math><mi>G</mi></math></span> and leaves every remaining component with at least <span><math><mi>h</mi></math></span> vertices. The <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-enhanced hypercubes <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> are constructed by adding a matching between some pair copies of <span><math><mi>k</mi></math></span> dimensional subcubes <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> with <span><math><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>. The distribution of the values of the <span><math><mi>h</mi></math></span>-extra edge-connectivity on a recursive graph is uneven and presents a concentration phenomenon. In this paper, we start with analysing the fractal properties of the optimal solution of the edge isoperimetric problem of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span>. And it is shown that although the members of <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>k</mi></mrow></msub></math></span> are not isomorphic to each other according to different <span><math><mi>k</mi></math></span> where <span><math><mrow><mn>2</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>, when <span><math><mi>n</mi></math></span> approaches infinity, the <span><math><mi>h</mi></math></span>-extra edge-connectivity of <span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-enhanced hypercubes presents a concentration phenome","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 167-174"},"PeriodicalIF":1.0,"publicationDate":"2025-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143734923","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 : 2025-03-29DOI: 10.1016/j.dam.2025.03.029
Brian Curcio, Isabel Méndez-Díaz, Paula Zabala
In this work we define the adjacent vertex distinguishing sum edge coloring problem. This problem consists of finding an assignment of colors to the edges of a graph with the following constraints: every pair of adjacent edges must have a different color, and every pair of adjacent vertices must not have the same set of colors assigned to the edges incident to each. The goal is to minimize the sum of the colors in an edge coloring that satisfies these constraints. This problem is a special case of a large family of problems known as graph labeling, which is a widely used and very popular set of tools to build abstract models for problems that arise in everyday life.
Some variants of graph labeling problems have been successfully addressed with mixed-integer linear programming (MIP) techniques based on a polyhedral characterization of the set of feasible solutions. We use this approach to develop a Branch and Cut algorithm to solve the problem.
We propose two MIP models that are computationally evaluated to choose the most promising one and continue with a polyhedral study. This analysis aims to characterize valid inequalities that strengthen the formulation in the hope of improving the algorithm’s performance. These inequalities are added on demand as cutting planes using exact and heuristic separation algorithms. Additionally, we considered the use of an initial heuristic and a specific branching strategy.
The results show that the algorithm developed allows us to solve instances that were unsolvable using general-purpose solvers. Our polyhedral study and the addition of cutting planes have proved to be crucial factors in solving the most challenging instances.
{"title":"An exact algorithm for the adjacent vertex distinguishing sum edge coloring problem","authors":"Brian Curcio, Isabel Méndez-Díaz, Paula Zabala","doi":"10.1016/j.dam.2025.03.029","DOIUrl":"10.1016/j.dam.2025.03.029","url":null,"abstract":"<div><div>In this work we define the <em>adjacent vertex distinguishing sum edge coloring problem</em>. This problem consists of finding an assignment of colors to the edges of a graph with the following constraints: every pair of adjacent edges must have a different color, and every pair of adjacent vertices must not have the same set of colors assigned to the edges incident to each. The goal is to minimize the sum of the colors in an edge coloring that satisfies these constraints. This problem is a special case of a large family of problems known as <em>graph labeling</em>, which is a widely used and very popular set of tools to build abstract models for problems that arise in everyday life.</div><div>Some variants of <em>graph labeling problems</em> have been successfully addressed with mixed-integer linear programming (MIP) techniques based on a polyhedral characterization of the set of feasible solutions. We use this approach to develop a <em>Branch and Cut</em> algorithm to solve the problem.</div><div>We propose two MIP models that are computationally evaluated to choose the most promising one and continue with a polyhedral study. This analysis aims to characterize valid inequalities that strengthen the formulation in the hope of improving the algorithm’s performance. These inequalities are added on demand as cutting planes using exact and heuristic separation algorithms. Additionally, we considered the use of an initial heuristic and a specific branching strategy.</div><div>The results show that the algorithm developed allows us to solve instances that were unsolvable using general-purpose solvers. Our polyhedral study and the addition of cutting planes have proved to be crucial factors in solving the most challenging instances.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 80-98"},"PeriodicalIF":1.0,"publicationDate":"2025-03-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143734795","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 : 2025-03-28DOI: 10.1016/j.dam.2025.03.030
Dazhi Lin , Tao Wang
Recently, Lin introduced a new graph parameter defined by nine properties and established the inequality , where is the harmonic index of . In this note, we simplify the framework by reducing the required properties to five, defining a new parameter . We prove , where is the modified Randić index, and characterize the equality case. It is known that . Then . Since needs fewer properties than , it has a high possibility that many known parameters satisfy the demands of .
{"title":"A simplified graph parameter and its relationship to the modified Randić index","authors":"Dazhi Lin , Tao Wang","doi":"10.1016/j.dam.2025.03.030","DOIUrl":"10.1016/j.dam.2025.03.030","url":null,"abstract":"<div><div>Recently, Lin introduced a new graph parameter <span><math><mi>ξ</mi></math></span> defined by nine properties and established the inequality <span><math><mrow><mi>ξ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>H</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><mi>H</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the harmonic index of <span><math><mi>G</mi></math></span>. In this note, we simplify the framework by reducing the required properties to five, defining a new parameter <span><math><mi>ζ</mi></math></span>. We prove <span><math><mrow><mi>ζ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, where <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is the modified Randić index, and characterize the equality case. It is known that <span><math><mrow><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>H</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Then <span><math><mrow><mi>ζ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><msup><mrow><mi>R</mi></mrow><mrow><mo>′</mo></mrow></msup><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>H</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>2</mn><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Since <span><math><mi>ζ</mi></math></span> needs fewer properties than <span><math><mi>ξ</mi></math></span>, it has a high possibility that many known parameters satisfy the demands of <span><math><mi>ζ</mi></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 60-64"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143715403","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 : 2025-03-28DOI: 10.1016/j.dam.2025.03.031
Myungho Choi
We say that a digraph is -step competitive if any two vertices have an -step common out-neighbor in and that a graph is -step competitively orientable if there exists an -step competitive orientation of .
In Choi et al. (2022), Choi et al. introduce the notion of the competitive digraph and completely characterize competitively orientable complete multipartite graphs in terms of the sizes of its partite sets. Here, a competitive digraph means a -step competitive digraph. In this paper, the result of Choi et al. has been extended to a general characterization of -step competitively orientable complete multipartite graphs.
{"title":"Generalized competitively orientable complete multipartite graphs","authors":"Myungho Choi","doi":"10.1016/j.dam.2025.03.031","DOIUrl":"10.1016/j.dam.2025.03.031","url":null,"abstract":"<div><div>We say that a digraph <span><math><mi>D</mi></math></span> is <span><math><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></math></span>-step competitive if any two vertices have an <span><math><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></math></span>-step common out-neighbor in <span><math><mi>D</mi></math></span> and that a graph <span><math><mi>G</mi></math></span> is <span><math><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></math></span>-step competitively orientable if there exists an <span><math><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></math></span>-step competitive orientation of <span><math><mi>G</mi></math></span>.</div><div>In Choi et al. (2022), Choi et al. introduce the notion of the competitive digraph and completely characterize competitively orientable complete multipartite graphs in terms of the sizes of its partite sets. Here, a competitive digraph means a <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-step competitive digraph. In this paper, the result of Choi et al. has been extended to a general characterization of <span><math><mrow><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></mrow></math></span>-step competitively orientable complete multipartite graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 65-72"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143715404","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 : 2025-03-28DOI: 10.1016/j.dam.2025.03.025
David Ashok Kalarkop , Michael A. Henning , Ismail Sahul Hamid , Pawaton Kaemawichanurat
An irredundance coloring of a graph is a proper coloring admitting a maximal irredundant set all of whose vertices receive different colors. The minimum number of colors required for an irredundance coloring of is called the irredundance chromatic number of , and is denoted by . An irredundance compelling coloring of is a proper coloring of in which every rainbow committee (a set consisting of one vertex of each color) is an irredundant set of . The maximum number of colors required for an irredundance compelling coloring of is called the irredundance compelling chromatic number of , and is denoted by . We make a detailed study of , , derive bounds on these parameters and characterize extremal graphs attaining the bounds.
{"title":"On irredundance coloring and irredundance compelling coloring of graphs","authors":"David Ashok Kalarkop , Michael A. Henning , Ismail Sahul Hamid , Pawaton Kaemawichanurat","doi":"10.1016/j.dam.2025.03.025","DOIUrl":"10.1016/j.dam.2025.03.025","url":null,"abstract":"<div><div>An irredundance coloring of a graph <span><math><mi>G</mi></math></span> is a proper coloring admitting a maximal irredundant set all of whose vertices receive different colors. The minimum number of colors required for an irredundance coloring of <span><math><mi>G</mi></math></span> is called the <em>irredundance chromatic number</em> of <span><math><mi>G</mi></math></span>, and is denoted by <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. An irredundance compelling coloring of <span><math><mi>G</mi></math></span> is a proper coloring of <span><math><mi>G</mi></math></span> in which every rainbow committee (a set consisting of one vertex of each color) is an irredundant set of <span><math><mi>G</mi></math></span>. The maximum number of colors required for an irredundance compelling coloring of <span><math><mi>G</mi></math></span> is called the <em>irredundance compelling chromatic number</em> of <span><math><mi>G</mi></math></span>, and is denoted by <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>i</mi><mi>r</mi><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. We make a detailed study of <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>i</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, <span><math><mrow><msub><mrow><mi>χ</mi></mrow><mrow><mi>i</mi><mi>r</mi><mi>c</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, derive bounds on these parameters and characterize extremal graphs attaining the bounds.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"369 ","pages":"Pages 149-161"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143715628","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2025-03-28DOI: 10.1016/j.dam.2025.03.012
Anand Brahmbhatt , Kartikeya Rai , Amitabha Tripathi
A graph is cordial if there exists a function from the vertices of to such that the number of vertices labelled 0 and the number of vertices labelled 1 differ by at most 1, and if we assign to each edge the label , the number of edges labelled 0 and the number of edges labelled 1 also differ at most by 1. We introduce two measures of how close a graph is to being cordial, and compute these measures for a variety of classes of graphs.
{"title":"Measures of closeness to cordiality for graphs","authors":"Anand Brahmbhatt , Kartikeya Rai , Amitabha Tripathi","doi":"10.1016/j.dam.2025.03.012","DOIUrl":"10.1016/j.dam.2025.03.012","url":null,"abstract":"<div><div>A graph <span><math><mi>G</mi></math></span> is cordial if there exists a function <span><math><mi>f</mi></math></span> from the vertices of <span><math><mi>G</mi></math></span> to <span><math><mrow><mo>{</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>}</mo></mrow></math></span> such that the number of vertices labelled 0 and the number of vertices labelled 1 differ by at most 1, and if we assign to each edge <span><math><mrow><mi>x</mi><mi>y</mi></mrow></math></span> the label <span><math><mrow><mo>|</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>−</mo><mi>f</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>|</mo></mrow></math></span>, the number of edges labelled 0 and the number of edges labelled 1 also differ at most by 1. We introduce two measures of how close a graph is to being cordial, and compute these measures for a variety of classes of graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"370 ","pages":"Pages 157-166"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143715551","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 : 2025-03-28DOI: 10.1016/j.dam.2025.03.022
Jiuying Dong, Yingying Yao
Let be a graph with vertices and edges. The energy of a graph is defined as the sum of absolute values of the eigenvalues about its adjacency matrix, i.e. . In this paper, we derive some new upper bounds on the graph energy based on a new formula and some inequalities for calculating the graph energy, and characterize the extremal graphs. In addition, we propose some new lower bounds for the graph energy involving order , the size , the eigenvalue with maximum absolute value and the eigenvalue with minimum absolute value of the graph , and characterize the extremal graphs.
{"title":"Some new bounds for the energy of graphs","authors":"Jiuying Dong, Yingying Yao","doi":"10.1016/j.dam.2025.03.022","DOIUrl":"10.1016/j.dam.2025.03.022","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a graph with <span><math><mi>n</mi></math></span> vertices and <span><math><mi>m</mi></math></span> edges. The energy of a graph <span><math><mi>G</mi></math></span> is defined as the sum of absolute values of the eigenvalues about its adjacency matrix, i.e. <span><math><mrow><mi>ɛ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mrow><mo>|</mo><msub><mrow><mi>λ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>|</mo></mrow></mrow></math></span>. In this paper, we derive some new upper bounds on the graph energy based on a new formula and some inequalities for calculating the graph energy, and characterize the extremal graphs. In addition, we propose some new lower bounds for the graph energy involving order <span><math><mi>n</mi></math></span>, the size <span><math><mi>m</mi></math></span>, the eigenvalue with maximum absolute value <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and the eigenvalue with minimum absolute value <span><math><msub><mrow><mi>λ</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of the graph <span><math><mi>G</mi></math></span>, and characterize the extremal graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 73-79"},"PeriodicalIF":1.0,"publicationDate":"2025-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143715514","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 : 2025-03-25DOI: 10.1016/j.dam.2025.03.023
Manuel Lafond, Weidong Luo
Structural graph parameters play an important role in parameterized complexity, including in kernelization. Notably, vertex cover, neighborhood diversity, twin-cover, and modular-width have been studied extensively in the last few years. However, there are many fundamental problems whose preprocessing complexity is not fully understood under these parameters. Indeed, the existence of polynomial kernels or polynomial Turing kernels for famous problems such as Clique, Chromatic Number, and Steiner Tree has only been established for a subset of structural parameters. In this work, we use several techniques to obtain a complete preprocessing complexity landscape for over a dozen of fundamental algorithmic problems.
{"title":"Preprocessing complexity for some graph problems parameterized by structural parameters","authors":"Manuel Lafond, Weidong Luo","doi":"10.1016/j.dam.2025.03.023","DOIUrl":"10.1016/j.dam.2025.03.023","url":null,"abstract":"<div><div>Structural graph parameters play an important role in parameterized complexity, including in kernelization. Notably, vertex cover, neighborhood diversity, twin-cover, and modular-width have been studied extensively in the last few years. However, there are many fundamental problems whose preprocessing complexity is not fully understood under these parameters. Indeed, the existence of polynomial kernels or polynomial Turing kernels for famous problems such as <span>Clique</span>, <span>Chromatic Number</span>, and <span>Steiner Tree</span> has only been established for a subset of structural parameters. In this work, we use several techniques to obtain a complete preprocessing complexity landscape for over a dozen of fundamental algorithmic problems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 46-59"},"PeriodicalIF":1.0,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696945","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 : 2025-03-25DOI: 10.1016/j.dam.2025.03.020
Peng-Li Zhang , Xiao-Dong Zhang
Let and be the diagonal and adjacency tensors of a -uniform hypergraph respectively. The -spectral radius of is defined as the spectral radius of the tensor where In this paper, we obtain an interlacing inequality on the spectral radius of a principal subtensor for a nonnegative weakly irreducible symmetric tensor, which is used to present several sharp lower bounds for the -spectral radius of any subhypergraph of a connected -uniform hypergraph in terms of the principal eigenvector associated with the -spectral radius of , degrees and co-degrees, where is a subset of . They extend and strengthen some known results.
{"title":"Bounds on the Aα-spectral radius of uniform hypergraphs with some vertices deleted","authors":"Peng-Li Zhang , Xiao-Dong Zhang","doi":"10.1016/j.dam.2025.03.020","DOIUrl":"10.1016/j.dam.2025.03.020","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the diagonal and adjacency tensors of a <span><math><mi>k</mi></math></span>-uniform hypergraph <span><math><mrow><mi>G</mi><mo>,</mo></mrow></math></span> respectively. The <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of <span><math><mi>G</mi></math></span> is defined as the spectral radius of the tensor <span><math><mrow><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>α</mi><mi>D</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>α</mi><mo>)</mo></mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>,</mo></mrow></math></span> where <span><math><mrow><mn>0</mn><mo>≤</mo><mi>α</mi><mo><</mo><mn>1</mn><mo>.</mo></mrow></math></span> In this paper, we obtain an interlacing inequality on the spectral radius of a principal subtensor for a nonnegative weakly irreducible symmetric tensor, which is used to present several sharp lower bounds for the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of any subhypergraph <span><math><mrow><mi>G</mi><mo>−</mo><mi>S</mi></mrow></math></span> of a connected <span><math><mi>k</mi></math></span>-uniform hypergraph <span><math><mi>G</mi></math></span> in terms of the principal eigenvector associated with the <span><math><msub><mrow><mi>A</mi></mrow><mrow><mi>α</mi></mrow></msub></math></span>-spectral radius of <span><math><mi>G</mi></math></span>, degrees and co-degrees, where <span><math><mi>S</mi></math></span> is a subset of <span><math><mrow><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. They extend and strengthen some known results.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"371 ","pages":"Pages 1-16"},"PeriodicalIF":1.0,"publicationDate":"2025-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143696939","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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