Discrete Mathematics最新文献_第4页

Induced matching vs edge open packing: Trees and product graphs

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-04 DOI: 10.1016/j.disc.2025.114458

Boštjan Brešar , Tanja Dravec , Jaka Hedžet , Babak Samadi

{"title":"Induced matching vs edge open packing: Trees and product graphs","authors":"Boštjan Brešar , Tanja Dravec , Jaka Hedžet , Babak Samadi","doi":"10.1016/j.disc.2025.114458","DOIUrl":"10.1016/j.disc.2025.114458","url":null,"abstract":"<div><div>Given a graph G, the maximum size of an induced subgraph of G each component of which is a star is called the edge open packing number, <math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo></math>, of G. Similarly, the maximum size of an induced subgraph of G each component of which is the star <math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mn>1</mn></mrow></msub></math> is the induced matching number, <math><msub><mrow><mi>ν</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math>, of G. While the inequality <math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math> clearly holds for all graphs G, we provide a structural characterization of those trees that attain the equality. We prove that the induced matching number of the lexicographic product <math><mi>G</mi><mo>∘</mo><mi>H</mi></math> of arbitrary two graphs G and H equals <math><mi>α</mi><mo>(</mo><mi>G</mi><mo>)</mo><msub><mrow><mi>ν</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>H</mi><mo>)</mo></math>. By similar techniques, we prove sharp lower and upper bounds on the edge open packing number of the lexicographic product of graphs, which in particular lead to NP-hardness results in triangular graphs for both invariants studied in this paper. For the direct product <math><mi>G</mi><mo>×</mo><mi>H</mi></math> of two graphs we provide lower bounds on <math><msub><mrow><mi>ν</mi></mrow><mrow><mi>I</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>×</mo><mi>H</mi><mo>)</mo></math> and <math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><mi>G</mi><mo>×</mo><mi>H</mi><mo>)</mo></math>, both of which are widely sharp. We also present sharp lower bounds for both invariants in the Cartesian and the strong product of two graphs. Finally, we consider the edge open packing number in hypercubes establishing the exact values of <math><msubsup><mrow><mi>ρ</mi></mrow><mrow><mi>e</mi></mrow><mrow><mi>o</mi></mrow></msubsup><mo>(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math> when n is a power of 2, and present a closed formula for the induced matching number of the rooted product of arbitrary two graphs over an arbitrary root vertex.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114458"},"PeriodicalIF":0.7,"publicationDate":"2025-03-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143535043","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}

引用次数: 0

On HV-neighborhood group constant sum array

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-03 DOI: 10.1016/j.disc.2025.114456

Karthik S, Krishnan Paramasivam

{"title":"On HV-neighborhood group constant sum array","authors":"Karthik S, Krishnan Paramasivam","doi":"10.1016/j.disc.2025.114456","DOIUrl":"10.1016/j.disc.2025.114456","url":null,"abstract":"<div><div>A HV-neighborhood group constant sum array with δ distance, is an <math><mi>m</mi><mo>×</mo><mi>n</mi></math> array, whose entries are all non-zero elements of an additive Abelian group Γ such that the sum of group elements assigned to the δ-neighborhood of any cell <math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math> in an <math><mi>m</mi><mo>×</mo><mi>n</mi></math> array is a unique element <math><mi>μ</mi><mo>∈</mo><mi>Γ</mi></math>, where the δ-neighborhood of a cell <math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math> in an <math><mi>m</mi><mo>×</mo><mi>n</mi></math> array is the set of cells that are at most δ distance in the right, left, up, and down from <math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math>, excluding the cell <math><mo>(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo>)</mo></math>. The element μ is the neighborhood constant sum. In this article, we prove some necessary conditions for the existence of HV-neighborhood group constant sum arrays with δ distance. In addition, if <math><mi>δ</mi><mo>=</mo><mn>1</mn></math>, a method to construct HV-neighborhood group constant sum arrays and a characterization of HV-neighborhood Klein four-group constant sum arrays are given.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114456"},"PeriodicalIF":0.7,"publicationDate":"2025-03-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143535039","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}

引用次数: 0

F2F2[u2]F2[u3]-additive cyclic codes are asymptotically good

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-03-03 DOI: 10.1016/j.disc.2025.114459

Hai Q. Dinh , Bhanu Pratap Yadav , Bac T. Nguyen , Ashish Kumar Upadhyay

引用次数: 0

Path eccentricity of k-AT-free graphs and application on graphs with the consecutive ones property

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-02-28 DOI: 10.1016/j.disc.2025.114449

Paul Bastide , Claire Hilaire , Eileen Robinson

{"title":"Path eccentricity of k-AT-free graphs and application on graphs with the consecutive ones property","authors":"Paul Bastide , Claire Hilaire , Eileen Robinson","doi":"10.1016/j.disc.2025.114449","DOIUrl":"10.1016/j.disc.2025.114449","url":null,"abstract":"<div><div>The central path problem is a variation on the single facility location problem. The aim is to find, in a given connected graph G, a path P minimizing its eccentricity, which is the maximal distance from P to any vertex of the graph G. The path eccentricity of G is the minimal eccentricity achievable over all paths in G. In this article we consider the path eccentricity of the class of the k-AT-free graphs. They are graphs in which any set of three vertices contains a pair for which every path between them uses at least one vertex of the closed neighborhood at distance k of the third. We prove that they have path eccentricity bounded by k.</div><div>Moreover, we answer a question of Gómez and Gutiérrez asking if there is a relation between path eccentricity and the consecutive ones property. The latter is the property for a binary matrix to admit a permutation of the rows placing the 1's consecutively on the columns. It was already known that graphs whose adjacency matrices have the consecutive ones property have path eccentricity at most 1, and that the same remains true when the augmented adjacency matrices (with ones on the diagonal) have the consecutive ones property. We generalize these results as follow. We study graphs whose adjacency matrices can be made to satisfy the consecutive ones property after changing some values on the diagonal, and show that those graphs have path eccentricity at most 2, by showing that they are 2-AT-free.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114449"},"PeriodicalIF":0.7,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511942","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}

引用次数: 0

On intersecting families of subgraphs of perfect matchings

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-02-28 DOI: 10.1016/j.disc.2025.114460

Melissa Fuentes, Vikram Kamat

{"title":"On intersecting families of subgraphs of perfect matchings","authors":"Melissa Fuentes, Vikram Kamat","doi":"10.1016/j.disc.2025.114460","DOIUrl":"10.1016/j.disc.2025.114460","url":null,"abstract":"<div><div>The seminal Erdős–Ko–Rado (EKR) theorem states that if <math><mi>F</mi></math> is a family of k-subsets of an n-element set X for <math><mi>k</mi><mo>≤</mo><mi>n</mi><mo>/</mo><mn>2</mn></math> such that every pair of subsets in <math><mi>F</mi></math> has a nonempty intersection, then <math><mi>F</mi></math> can be no bigger than the trivially intersecting family obtained by including all k-subsets of X that contain a fixed element <math><mi>x</mi><mo>∈</mo><mi>X</mi></math>. This family is called the star centered at x. In this paper, we formulate and prove an EKR theorem for intersecting families of subgraphs of the perfect matching graph. This can be considered a generalization not only of the aforementioned EKR theorem but also of a signed variant of it, first stated by Meyer [9], and proved separately by Deza–Frankl [3] and Bollobás–Leader [1]. The proof of our main theorem relies on a novel extension of Katona's beautiful cycle method.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 7","pages":"Article 114460"},"PeriodicalIF":0.7,"publicationDate":"2025-02-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143511944","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}

引用次数: 0

Weighted Padovan graphs

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-02-28 DOI: 10.1016/j.disc.2025.114457

Vesna Iršič Chenoweth , Sandi Klavžar , Gregor Rus , Elif Tan

引用次数: 0

Remoteness of graphs with given size and connectivity constraints

IF 0.7 3区数学

Discrete Mathematics Pub Date : 2025-02-26 DOI: 10.1016/j.disc.2025.114451

Peter Dankelmann , Sonwabile Mafunda , Sufiyan Mallu

引用次数: 0

Poset positional games