{"title":"Degree sequence optimization and extremal degree enumerators","authors":"Shmuel Onn","doi":"10.1016/j.dam.2025.02.008","DOIUrl":"10.1016/j.dam.2025.02.008","url":null,"abstract":"<div><div>The degree sequence optimization problem is to find a subgraph of a given graph which maximizes the sum of given functions evaluated at the subgraph degrees. Here we study this problem by replacing degree sequences, via suitable nonlinear transformations, by suitable degree enumerators, and we introduce suitable degree enumerator polytopes.</div><div>We characterize their vertices, that is, the extremal degree enumerators, for complete graphs and some complete bipartite graphs, and use these characterizations to obtain simpler and faster algorithms for optimization over degree sequences for such graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 107-115"},"PeriodicalIF":1.0,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143387167","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}
{"title":"An extended hypergraph cut method for the Wiener index","authors":"Gašper Domen Romih","doi":"10.1016/j.dam.2025.02.003","DOIUrl":"10.1016/j.dam.2025.02.003","url":null,"abstract":"<div><div>The cut method for the Wiener index of <span><math><mi>k</mi></math></span>-uniform partial cube hypergraphs was recently proposed. This paper introduces a method for the non-uniform case. The method is also extended to some non-uniform and non-linear hypergraphs which are not partial cube hypergraphs. The extended method is applied to cube hypergraphs, hypertrees and phenylene hypergraphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 80-88"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378804","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}
{"title":"On the smallest positive eigenvalue of caterpillar unicyclic graphs","authors":"Sasmita Barik, Subhasish Behera","doi":"10.1016/j.dam.2025.02.002","DOIUrl":"10.1016/j.dam.2025.02.002","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a simple graph with the adjacency matrix <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. By the smallest positive eigenvalue of <span><math><mi>G</mi></math></span>, we mean the smallest positive eigenvalue of <span><math><mrow><mi>A</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and denote it by <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. For <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be the cycle graph with vertices <span><math><mrow><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></math></span> be <span><math><mi>k</mi></math></span> nonnegative integers. A caterpillar unicyclic graph <span><math><mrow><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub><mrow><mo>(</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> is a graph obtained from <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> by adding <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> pendant vertices to the vertex <span><math><mi>i</mi></math></span> of <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for <span><math><mrow><mi>i</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi></mrow></math></span>. Let <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be the class of all caterpillar unicyclic graphs on <span><math><mi>n</mi></math></span> vertices, where each <span><math><msub><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is positive. In this article, we obtain the graphs with the maximum <span><math><mi>τ</mi></math></span> among all the graphs in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. Furthermore, we characterize the graphs <span><math><mi>G</mi></math></span> in <span><math><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> such that <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>></mo><msqrt><mrow><mn>2</mn></mrow></msqrt><mo>−</mo><","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 89-98"},"PeriodicalIF":1.0,"publicationDate":"2025-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143378805","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}
Wenjuan Zhou , Rong-Xia Hao , Rong Luo , Cun-Quan Zhang
{"title":"4-coverable snarks, perfect matching cover, and Isaacs product","authors":"Wenjuan Zhou , Rong-Xia Hao , Rong Luo , Cun-Quan Zhang","doi":"10.1016/j.dam.2025.02.001","DOIUrl":"10.1016/j.dam.2025.02.001","url":null,"abstract":"<div><div>The perfect matching cover index <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> of a bridgeless cubic graph <span><math><mi>G</mi></math></span> is the minimum number of perfect matchings in <span><math><mi>G</mi></math></span> that cover all edges of <span><math><mi>G</mi></math></span>. It is conjectured by Berge that <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>5</mn></mrow></math></span> for every bridgeless cubic graph. Esperet and Mazzuoccolo (2013) proved that deciding whether a cubic bridgeless graph <span><math><mi>G</mi></math></span> satisfies <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≤</mo><mn>4</mn></mrow></math></span> is NP-complete. Some major conjectures hold for bridgeless cubic graphs <span><math><mi>G</mi></math></span> with <span><math><mrow><mi>τ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mn>4</mn></mrow></math></span>, such as the cycle double cover conjecture and the Alon–Tarsi shortest cycle cover conjecture. In this paper, we study the perfect matching cover index of snarks obtained by the Isaacs products of two cubic graphs. Our results indicate that the Isaacs product of certain snarks does not increase the perfect matching cover index. As corollaries, we show that the perfect matching cover indices of some families of snarks with arbitrarily large oddness equal to 4.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 68-79"},"PeriodicalIF":1.0,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349902","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}
Fiorenza Morini , Marco Antonio Pellegrini , Stefania Sora
{"title":"On a conjecture by Sylwia Cichacz and Tomasz Hinc, and a related problem","authors":"Fiorenza Morini , Marco Antonio Pellegrini , Stefania Sora","doi":"10.1016/j.dam.2025.01.040","DOIUrl":"10.1016/j.dam.2025.01.040","url":null,"abstract":"<div><div>A <span><math><mi>Γ</mi></math></span>-magic rectangle set <span><math><mrow><msub><mrow><mi>MRS</mi></mrow><mrow><mi>Γ</mi></mrow></msub><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>)</mo></mrow></mrow></math></span> is a collection of <span><math><mi>c</mi></math></span> arrays of size <span><math><mrow><mi>a</mi><mo>×</mo><mi>b</mi></mrow></math></span> whose entries are the elements of an abelian group <span><math><mi>Γ</mi></math></span> of order <span><math><mrow><mi>a</mi><mi>b</mi><mi>c</mi></mrow></math></span>, each one appearing once and in a unique array in such a way that the sum of the entries of each row is equal to a constant <span><math><mrow><mi>ω</mi><mo>∈</mo><mi>Γ</mi></mrow></math></span> and the sum of the entries of each column is equal to a constant <span><math><mrow><mi>δ</mi><mo>∈</mo><mi>Γ</mi></mrow></math></span>.</div><div>In this paper we provide new evidences for the validity of a conjecture proposed by Sylwia Cichacz and Tomasz Hinc on the existence of an <span><math><mrow><msub><mrow><mi>MRS</mi></mrow><mrow><mi>Γ</mi></mrow></msub><mrow><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>;</mo><mi>c</mi><mo>)</mo></mrow></mrow></math></span>. We also generalize this problem describing constructions of <span><math><mi>Γ</mi></math></span>-magic rectangle sets whose elements are partially filled arrays.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 53-67"},"PeriodicalIF":1.0,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143348923","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}
{"title":"The number of spanning trees in Km,n-complements of bipartite graphs","authors":"Helin Gong , Xiurong Yan , Anshui Li","doi":"10.1016/j.dam.2025.01.039","DOIUrl":"10.1016/j.dam.2025.01.039","url":null,"abstract":"<div><div>For a subgraph <span><math><mi>G</mi></math></span> in the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>, the <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>-complement of <span><math><mi>G</mi></math></span> (namely <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>−</mo><mi>G</mi></mrow></math></span>) is the graph obtained from <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span> by removing the edges of <span><math><mi>G</mi></math></span>. In this paper, by the matrix-tree theorem, we derive a general expression for the number of spanning trees of <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub><mo>−</mo><mi>G</mi></mrow></math></span> and establish explicit formulas for the number of spanning trees of the <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>-complements of various classes of bipartite graphs, which generalizes some known results.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 40-52"},"PeriodicalIF":1.0,"publicationDate":"2025-02-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143349903","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}
{"title":"Total list weighting of Cartesian product of graphs","authors":"Yunfang Tang , Yuting Yao","doi":"10.1016/j.dam.2025.01.043","DOIUrl":"10.1016/j.dam.2025.01.043","url":null,"abstract":"<div><div>A proper total weighting of a graph <span><math><mi>G</mi></math></span> is a mapping <span><math><mi>ϕ</mi></math></span> that assigns a real number as the weight to each vertex and each edge of <span><math><mi>G</mi></math></span> so that for any two adjacent vertices <span><math><mi>u</mi></math></span> and <span><math><mi>v</mi></math></span>, <span><math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow></mrow></msub><mi>ϕ</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>+</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>u</mi><mo>)</mo></mrow><mo>≠</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>e</mi><mo>∈</mo><mi>E</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></msub><mi>ϕ</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow><mo>+</mo><mi>ϕ</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span>. A graph <span><math><mrow><mi>G</mi><mo>=</mo><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></mrow></mrow></math></span> is called <span><math><mrow><mo>(</mo><mi>k</mi><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>)</mo></mrow></math></span>-choosable if the following is true: If each vertex <span><math><mi>v</mi></math></span> is assigned a set <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>v</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><mi>k</mi></math></span> real numbers, and each edge <span><math><mi>e</mi></math></span> is assigned a set <span><math><mrow><mi>L</mi><mrow><mo>(</mo><mi>e</mi><mo>)</mo></mrow></mrow></math></span> of <span><math><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> real numbers, then there is a proper total weighting <span><math><mi>ϕ</mi></math></span> with <span><math><mrow><mi>ϕ</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow><mo>∈</mo><mi>L</mi><mrow><mo>(</mo><mi>z</mi><mo>)</mo></mrow></mrow></math></span> for any <span><math><mrow><mi>z</mi><mo>∈</mo><mi>V</mi><mo>∪</mo><mi>E</mi></mrow></math></span>. In this paper, we prove that if <span><math><mi>G</mi></math></span> is the Cartesian product of a path and a cycle or the Cartesian product of two cycles, then <span><math><mi>G</mi></math></span> is <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow></math></span>-choosable and <span><math><mrow><mo>(</mo><mn>2</mn><mo>,</mo><mn>2</mn><mo>)</mo></mrow></math></span>-choosable. This improves and extends the known results which were proved by Wong et al. in 2012.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 30-39"},"PeriodicalIF":1.0,"publicationDate":"2025-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143225982","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}
{"title":"Solving the problem about the second largest normalized Laplacian eigenvalue","authors":"Xinghui Zhao, Lihua You","doi":"10.1016/j.dam.2025.01.041","DOIUrl":"10.1016/j.dam.2025.01.041","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a simple graph of order <span><math><mi>n</mi></math></span> with normalized Laplacian eigenvalue <span><math><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>ρ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>=</mo><mn>0</mn></mrow></math></span>. In Sun and Das (2019), the authors obtained the first three smallest values on <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> of connected graphs and proposed an open problem on the third smallest values of <span><math><msub><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>. In this paper, we solve the problem completely, characterize all connected graphs with order <span><math><mi>n</mi></math></span>, which satisfy <span><math><mrow><msub><mrow><mi>ρ</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mfrac></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 8-21"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143168315","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}
{"title":"The phase transitions of diameters in random axis-parallel hyperrectangle intersection graphs","authors":"Congsong Zhang , Yong Gao , James Nastos","doi":"10.1016/j.dam.2025.01.044","DOIUrl":"10.1016/j.dam.2025.01.044","url":null,"abstract":"<div><div>We study the behaviours of diameters in two models of random axis-parallel hyperrectangle intersection graphs: <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>u</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>. These two models use axis-parallel <span><math><mi>l</mi></math></span>-dimensional hyperrectangles to represent vertices, and vertices are adjacent if and only if their corresponding axis-parallel hyperrectangles intersect. In the model <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>, we distribute <span><math><mi>n</mi></math></span> points within <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>l</mi></mrow></msup></math></span> uniformly and independently, and each point is the centre of an axis-parallel <span><math><mi>l</mi></math></span>-dimensional hypercube with edge length <span><math><mi>r</mi></math></span>. The model <span><math><mrow><msub><mrow><mi>G</mi></mrow><mrow><mi>u</mi></mrow></msub><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>, distributing the centres of <span><math><mi>n</mi></math></span> axis-parallel <span><math><mi>l</mi></math></span>-dimensional hyperrectangles within <span><math><msup><mrow><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></mrow><mrow><mi>l</mi></mrow></msup></math></span> exactly as the model <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>, assigns a length from a uniform distribution over <span><math><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mi>r</mi><mo>]</mo></mrow></math></span> to each edge of the <span><math><mi>n</mi></math></span> axis-parallel <span><math><mi>l</mi></math></span>-dimensional hyperrectangles.</div><div>We prove that in the model <span><math><mrow><mi>G</mi><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>,</mo><mi>l</mi><mo>)</mo></mrow></mrow></math></span>, there is a phase transition for the event that the diameter is at most <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></math></span> occurring at <span><math><mrow><mi>r</mi><mo>=</mo><mi>d</mi><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></mrow></math></span> if <span><math><mrow><mi>n</mi><mi>⋅</mi><mi>d</mi><msup><mrow><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow><mrow><mo>−</mo><mi>l</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></mrow></msup><mo>≥</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>ϵ</mi></mrow></msup><mo>,</mo></mrow></math></span>\u0000 where <span><math><mrow><mn>0</mn><mo><</mo><mi>ϵ</mi><","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"367 ","pages":"Pages 22-29"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143225980","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}
{"title":"Bounds for boxicity of circular clique graphs and zero-divisor graphs","authors":"T. Kavaskar","doi":"10.1016/j.dam.2025.01.038","DOIUrl":"10.1016/j.dam.2025.01.038","url":null,"abstract":"<div><div>Let <span><math><mrow><mi>b</mi><mi>o</mi><mi>x</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the boxicity of a graph <span><math><mi>G</mi></math></span>, <span><math><mrow><mi>G</mi><mrow><mo>[</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow></mrow></math></span> be the <span><math><mi>G</mi></math></span>-join graph of <span><math><mi>n</mi></math></span>-pairwise disjoint graphs <span><math><mrow><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, <span><math><msubsup><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msubsup></math></span> be a circular clique graph (where <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn><mi>d</mi></mrow></math></span>) and <span><math><mrow><mi>Γ</mi><mrow><mo>(</mo><mi>R</mi><mo>)</mo></mrow></mrow></math></span> be the zero-divisor graph of a commutative ring <span><math><mi>R</mi></math></span> with unity. In this paper, we prove that <span><math><mrow><mi>χ</mi><mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo></mrow><mo>≥</mo><mi>b</mi><mi>o</mi><mi>x</mi><mrow><mo>(</mo><msubsup><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow><mrow><mi>d</mi></mrow></msubsup><mo>)</mo></mrow></mrow></math></span>, for all <span><math><mi>k</mi></math></span> and <span><math><mi>d</mi></math></span> with <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn><mi>d</mi></mrow></math></span>. This generalizes the results proved by Kamibeppu (2018). Also we obtain that <span><math><mrow><mi>b</mi><mi>o</mi><mi>x</mi><mrow><mo>(</mo><mi>G</mi><mrow><mo>[</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>]</mo></mrow><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><mi>b</mi><mi>o</mi><mi>x</mi><mrow><mo>(</mo><msub><mrow><mi>H</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span>. As a consequence of this result, we obtain a bound for boxicity of ideal-based zero-divisor graph of a finite commutative ring with unity. In particular, if <span><math><mrow><mi>R</mi><mo>≇</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>×</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></mrow></math></span> is a finite commutative non-zero reduced ring with unity, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>q</mi></mrow></msub></math></sp","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"365 ","pages":"Pages 260-269"},"PeriodicalIF":1.0,"publicationDate":"2025-02-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143150139","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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