Pub Date : 2026-01-08DOI: 10.1016/j.dam.2025.12.065
Fernando Esteban Contreras-Mendoza, César Hernández-Cruz
A partition of the vertex set of a graph is called a polar partition of when and are complete multipartite graphs. If is a polar partition of in which and have at most and parts, respectively, then is an -polar partition of , and is said to be an -polar graph. A graph is said to be unipolar (monopolar) if its vertex set admits a polar partition such that is a clique (an independent set, respectively). A graph admitting a -polar partition is usually called a split graph.
Naturally, most problems related to polar partitions are trivial on split graphs, even when some of them are very hard in general. In this work, we present some results related to polar partitions on two graph classes generalizing split graphs. Our main results include efficient algorithms to decide whether graphs in these classes admit such partitions. We also establish upper bounds on the order of minimal -polar obstructions for these families, for any and (possibly or ).
{"title":"Polarity on H-split graphs","authors":"Fernando Esteban Contreras-Mendoza, César Hernández-Cruz","doi":"10.1016/j.dam.2025.12.065","DOIUrl":"10.1016/j.dam.2025.12.065","url":null,"abstract":"<div><div>A partition <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> of the vertex set of a graph <span><math><mi>G</mi></math></span> is called a polar partition of <span><math><mi>G</mi></math></span> when <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow></mrow></math></span> and <span><math><mover><mrow><mi>G</mi><mrow><mo>[</mo><mi>B</mi><mo>]</mo></mrow></mrow><mo>¯</mo></mover></math></span> are complete multipartite graphs. If <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> is a polar partition of <span><math><mi>G</mi></math></span> in which <span><math><mrow><mi>G</mi><mrow><mo>[</mo><mi>A</mi><mo>]</mo></mrow></mrow></math></span> and <span><math><mover><mrow><mi>G</mi><mrow><mo>[</mo><mi>B</mi><mo>]</mo></mrow></mrow><mo>¯</mo></mover></math></span> have at most <span><math><mi>s</mi></math></span> and <span><math><mi>k</mi></math></span> parts, respectively, then <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> is an <span><math><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-polar partition of <span><math><mi>G</mi></math></span>, and <span><math><mi>G</mi></math></span> is said to be an <span><math><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-polar graph. A graph is said to be unipolar (monopolar) if its vertex set admits a polar partition <span><math><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>)</mo></mrow></math></span> such that <span><math><mi>A</mi></math></span> is a clique (an independent set, respectively). A graph admitting a <span><math><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></math></span>-polar partition is usually called a split graph.</div><div>Naturally, most problems related to polar partitions are trivial on split graphs, even when some of them are very hard in general. In this work, we present some results related to polar partitions on two graph classes generalizing split graphs. Our main results include efficient algorithms to decide whether graphs in these classes admit such partitions. We also establish upper bounds on the order of minimal <span><math><mrow><mo>(</mo><mi>s</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></math></span>-polar obstructions for these families, for any <span><math><mi>s</mi></math></span> and <span><math><mi>k</mi></math></span> (possibly <span><math><mrow><mi>s</mi><mo>=</mo><mi>∞</mi></mrow></math></span> or <span><math><mrow><mi>k</mi><mo>=</mo><mi>∞</mi></mrow></math></span>).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 229-242"},"PeriodicalIF":1.0,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928984","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 : 2026-01-08DOI: 10.1016/j.dam.2025.12.055
Esteban De Jesús Sánchez , Abel Cabrera-Martínez , Ismael Rios-Villamar , José M. Sigarreta
In this article, we study the total domination number and the -domination number of the subdivision graph operator for simple graphs . First, we obtain a closed formula for the total domination number of this well-known graph operator. Additionally, for , we obtain the exact value of the -domination number of . The picture is notably different when considering the case , i.e., the domination number. In this case, we obtain some general bounds on the domination number of , which we further improve when is a tree. Finally, we provide closed formulas for the domination number of the subdivision graph of some well-known composite graphs.
{"title":"On the (total) domination in subdivision graphs","authors":"Esteban De Jesús Sánchez , Abel Cabrera-Martínez , Ismael Rios-Villamar , José M. Sigarreta","doi":"10.1016/j.dam.2025.12.055","DOIUrl":"10.1016/j.dam.2025.12.055","url":null,"abstract":"<div><div>In this article, we study the total domination number and the <span><math><mi>k</mi></math></span>-domination number of the subdivision graph operator <span><math><mrow><mstyle><mi>S</mi></mstyle><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for simple graphs <span><math><mi>G</mi></math></span>. First, we obtain a closed formula for the total domination number of this well-known graph operator. Additionally, for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>2</mn></mrow></math></span>, we obtain the exact value of the <span><math><mi>k</mi></math></span>-domination number of <span><math><mrow><mstyle><mi>S</mi></mstyle><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. The picture is notably different when considering the case <span><math><mrow><mi>k</mi><mo>=</mo><mn>1</mn></mrow></math></span>, i.e., the domination number. In this case, we obtain some general bounds on the domination number of <span><math><mrow><mstyle><mi>S</mi></mstyle><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>, which we further improve when <span><math><mi>G</mi></math></span> is a tree. Finally, we provide closed formulas for the domination number of the subdivision graph of some well-known composite graphs.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 201-210"},"PeriodicalIF":1.0,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928979","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 : 2026-01-08DOI: 10.1016/j.dam.2025.12.058
Taiki Yamada
The identification of vertices that play a central role in network analysis is a fundamental challenge. Although traditional centrality measures have been extensively employed for this purpose, the increasing complexity of modern networks necessitates the use of sophisticated metrics. The concept of Forman curvature has recently garnered significant attention as a promising approach. We define the Forman curvature for multiplex graphs, which are a category of complex networks characterized by multiple layers of connections between nodes. We then prove the key properties of the Forman curvature in the context of multiplex graphs and show its usefulness in identifying vertices occupying central positions within these networks. Moreover, through a series of comparative experiments with traditional graph features and graph kernels, we demonstrate that the Forman curvature can function as an effective metric for classifying the overall structure of networks.
{"title":"Vertex evaluation of multiplex graphs using Forman curvature","authors":"Taiki Yamada","doi":"10.1016/j.dam.2025.12.058","DOIUrl":"10.1016/j.dam.2025.12.058","url":null,"abstract":"<div><div>The identification of vertices that play a central role in network analysis is a fundamental challenge. Although traditional centrality measures have been extensively employed for this purpose, the increasing complexity of modern networks necessitates the use of sophisticated metrics. The concept of Forman curvature has recently garnered significant attention as a promising approach. We define the Forman curvature for multiplex graphs, which are a category of complex networks characterized by multiple layers of connections between nodes. We then prove the key properties of the Forman curvature in the context of multiplex graphs and show its usefulness in identifying vertices occupying central positions within these networks. Moreover, through a series of comparative experiments with traditional graph features and graph kernels, we demonstrate that the Forman curvature can function as an effective metric for classifying the overall structure of networks.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 211-228"},"PeriodicalIF":1.0,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928985","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 : 2026-01-08DOI: 10.1016/j.dam.2025.12.042
Martin Dyer , Haiko Müller
We consider classes of graphs, which we call thick graphs, that have the vertices of a corresponding thin graph replaced by cliques and the edges replaced by cobipartite graphs. In particular, we consider the case of thick forests, which we show to be the largest class of perfect thick graphs.
Recognising membership of a class of thick -graphs is NP-complete unless the class is triangle-free, so we focus on this case. Even then membership can be NP-complete. However, we show that the class of thick forests can be recognised in polynomial time.
We consider two well-studied combinatorial problems on thick graphs, independent sets and proper colourings. Since determining the independence or chromatic number of a perfect graph is known to be tractable, we examine the complexity of counting all independent sets and colourings in thick forests.
Finally, we consider two parametric extensions to larger classes of thick graphs: where the parameter is the size of the thin graph, and where the parameter is its treewidth.
{"title":"Thick Forests","authors":"Martin Dyer , Haiko Müller","doi":"10.1016/j.dam.2025.12.042","DOIUrl":"10.1016/j.dam.2025.12.042","url":null,"abstract":"<div><div>We consider classes of graphs, which we call <em>thick graphs</em>, that have the vertices of a corresponding <em>thin</em> graph replaced by cliques and the edges replaced by cobipartite graphs. In particular, we consider the case of <em>thick forests</em>, which we show to be the largest class of perfect thick graphs.</div><div>Recognising membership of a class of thick <span><math><mi>C</mi></math></span>-graphs is NP-complete unless the class <span><math><mi>C</mi></math></span> is triangle-free, so we focus on this case. Even then membership can be NP-complete. However, we show that the class of thick forests can be recognised in polynomial time.</div><div>We consider two well-studied combinatorial problems on thick graphs, independent sets and proper colourings. Since determining the independence or chromatic number of a perfect graph is known to be tractable, we examine the complexity of <em>counting</em> all independent sets and colourings in thick forests.</div><div>Finally, we consider two parametric extensions to larger classes of thick graphs: where the parameter is the size of the thin graph, and where the parameter is its treewidth.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 177-200"},"PeriodicalIF":1.0,"publicationDate":"2026-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928978","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 : 2026-01-07DOI: 10.1016/j.dam.2025.12.045
Pingshan Li, Ke Yang, Wei Jin
A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order and characterized the corresponding extremal graphs. They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order and the corresponding extremal graphs. In this paper, we give a positive answer to this question.
{"title":"A note on the second-largest number of dissociation sets in connected graphs","authors":"Pingshan Li, Ke Yang, Wei Jin","doi":"10.1016/j.dam.2025.12.045","DOIUrl":"10.1016/j.dam.2025.12.045","url":null,"abstract":"<div><div>A subset of vertices is called a dissociation set if it induces a subgraph with vertex degree at most one. Recently, Yuan et al. established the upper bound of the maximum number of dissociation sets among all connected graphs of order <span><math><mi>n</mi></math></span> and characterized the corresponding extremal graphs. They also proposed a question regarding the second-largest number of dissociation sets among all connected graphs of order <span><math><mi>n</mi></math></span> and the corresponding extremal graphs. In this paper, we give a positive answer to this question.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 34-40"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928977","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 : 2026-01-07DOI: 10.1016/j.dam.2025.12.046
Zi-Ming Zhou , Zhi-Bin Du , Chang-Xiang He
For a graph , let be the sum of the first two largest signless Laplacian eigenvalues of , and . Very recently, Zhou et al. (2024) proved that (the star graph with an additional edge) is the unique graph with minimum value of among the graphs on vertices. In this paper, we prove that the vertex-disjoint union of and possibly some isolated vertices is the unique graph with minimum value of among the graphs with edges.
{"title":"Extremal graphs for the sum of the first two largest signless Laplacian eigenvalues","authors":"Zi-Ming Zhou , Zhi-Bin Du , Chang-Xiang He","doi":"10.1016/j.dam.2025.12.046","DOIUrl":"10.1016/j.dam.2025.12.046","url":null,"abstract":"<div><div>For a graph <span><math><mi>G</mi></math></span>, let <span><math><mrow><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the sum of the first two largest signless Laplacian eigenvalues of <span><math><mi>G</mi></math></span>, and <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mn>3</mn><mo>−</mo><msub><mrow><mi>S</mi></mrow><mrow><mn>2</mn></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>. Very recently, Zhou et al. (2024) proved that <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> (the star graph with an additional edge) is the unique graph with minimum value of <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> among the graphs on <span><math><mi>n</mi></math></span> vertices. In this paper, we prove that the vertex-disjoint union of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mrow><mo>+</mo></mrow></msubsup></math></span> and possibly some isolated vertices is the unique graph with minimum value of <span><math><mrow><mi>f</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> among the graphs with <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> edges.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 139-144"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928987","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 : 2026-01-07DOI: 10.1016/j.dam.2025.12.051
Junqing Cai , Meirun Chen , Cheng-Kuan Lin
The hypercube and folded hypercube serve as fundamental interconnection network topologies in parallel computing, valued for their efficient communication and inherent fault tolerance. This paper investigates their resilience to dimensional-edge faults with respect to three critical Hamiltonian properties: Hamiltonicity, Hamiltonian laceability, and hyper Hamiltonian laceability. We establish precise bounds for fault tolerance in these structures, proving that: (1) For , both the dimensional-edge fault-tolerant Hamiltonicity and Hamiltonian laceability equal , while hyper Hamiltonian laceability tolerates up to ; (2) For , the dimensional-edge fault-tolerant Hamiltonicity is ; (3) For odd-dimensional , the dimensional-edge fault-tolerant Hamiltonian laceability and hyper Hamiltonian laceability are and , respectively. These results significantly advance our understanding of fault tolerance in cube-based network topologies and provide rigorous theoretical guarantees for their reliable operation in practical systems.
{"title":"Dimensional edge fault-tolerant Hamiltonicity of (folded) hypercubes","authors":"Junqing Cai , Meirun Chen , Cheng-Kuan Lin","doi":"10.1016/j.dam.2025.12.051","DOIUrl":"10.1016/j.dam.2025.12.051","url":null,"abstract":"<div><div>The hypercube <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> and folded hypercube <span><math><mrow><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span> serve as fundamental interconnection network topologies in parallel computing, valued for their efficient communication and inherent fault tolerance. This paper investigates their resilience to dimensional-edge faults with respect to three critical Hamiltonian properties: Hamiltonicity, Hamiltonian laceability, and hyper Hamiltonian laceability. We establish precise bounds for fault tolerance in these structures, proving that: (1) For <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, both the dimensional-edge fault-tolerant Hamiltonicity and Hamiltonian laceability equal <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mi>n</mi></mrow></math></span>, while hyper Hamiltonian laceability tolerates up to <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>n</mi><mo>+</mo><mn>2</mn></mrow></math></span>; (2) For <span><math><mrow><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow></math></span>, the dimensional-edge fault-tolerant Hamiltonicity is <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>−</mo><mi>n</mi></mrow></math></span>; (3) For odd-dimensional <span><math><mrow><mi>F</mi><msub><mrow><mi>Q</mi></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></math></span>, the dimensional-edge fault-tolerant Hamiltonian laceability and hyper Hamiltonian laceability are <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>2</mn><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span> and <span><math><mrow><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>−</mo><mn>4</mn><mi>n</mi></mrow></math></span>, respectively. These results significantly advance our understanding of fault tolerance in cube-based network topologies and provide rigorous theoretical guarantees for their reliable operation in practical systems.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 154-164"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928893","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 : 2026-01-07DOI: 10.1016/j.dam.2025.12.048
Yahui Zhang, Yan Li, Changxiang He
For graphs , and , let signify that any red-blue edge-coloring of contains either a red or a blue . The Ramsey number is defined as . In this note, we show that and under some bounds on , where and is a given graph with minimum color class of size 1. Our construction of edge-deleted balanced complete bipartite graphs removes more edges than earlier results concerning the critical Ramsey number of paths.
对于图F， G和H，设F→（G，H）表示F的任何红蓝边着色包含一个红色G或一个蓝色H，拉姆齐数R（G，H）定义为min{R |Kr→（G，H）}。在这篇文章中，我们证明了Kr∈Pr→（G,Pn）和Kr∈K≤≤n2²−1，≤n2²−1→（G,Pn）在n上的某些界下，其中r= r (G,Pn)，并且G是一个最小色类大小为1的给定图。我们构造的边删除平衡完全二部图比先前关于路径临界拉姆齐数的结果删除了更多的边。
{"title":"On some critical Ramsey numbers involving paths","authors":"Yahui Zhang, Yan Li, Changxiang He","doi":"10.1016/j.dam.2025.12.048","DOIUrl":"10.1016/j.dam.2025.12.048","url":null,"abstract":"<div><div>For graphs <span><math><mi>F</mi></math></span>, <span><math><mi>G</mi></math></span> and <span><math><mi>H</mi></math></span>, let <span><math><mrow><mi>F</mi><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> signify that any red-blue edge-coloring of <span><math><mi>F</mi></math></span> contains either a red <span><math><mi>G</mi></math></span> or a blue <span><math><mi>H</mi></math></span>. The Ramsey number <span><math><mrow><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow></mrow></math></span> is defined as <span><math><mrow><mi>m</mi><mi>i</mi><mi>n</mi><mrow><mo>{</mo><mi>r</mi><mspace></mspace><mo>|</mo><mspace></mspace><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span>. In this note, we show that <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>∖</mo><msub><mrow><mi>K</mi></mrow><mrow><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn><mo>,</mo><mrow><mo>⌈</mo><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac><mo>⌉</mo></mrow><mo>−</mo><mn>1</mn></mrow></msub><mo>→</mo><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> under some bounds on <span><math><mi>n</mi></math></span>, where <span><math><mrow><mi>r</mi><mo>=</mo><mi>R</mi><mrow><mo>(</mo><mi>G</mi><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></mrow></mrow></math></span> and <span><math><mi>G</mi></math></span> is a given graph with minimum color class of size 1. Our construction of edge-deleted balanced complete bipartite graphs removes more edges than earlier results concerning the critical Ramsey number of paths.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 145-153"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928982","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 : 2026-01-07DOI: 10.1016/j.dam.2025.12.053
Ming Jiang , Xiaogang Liu , Jing Wang
Quantum state transfer, first introduced by Bose in 2003, is an important physical phenomenon in quantum networks, which plays a vital role in quantum communication and quantum computing. In 2004, Christandl et al. proposed the concept of perfect state transfer on graphs by modeling the quantum network using graphs, and unveiled the feasibility of applying graph theory to quantum state transfer. In 2018, Chen and Godsil proposed the definition of Laplacian perfect pair state transfer on graphs, which is a brilliant generalization of perfect state transfer. In this paper, we investigate the existence of Laplacian perfect pair state transfer in tensor product and double cover of two regular graphs, respectively, and reveal fundamental connections between perfect state transfer and Laplacian perfect pair state transfer. We give necessary and sufficient conditions for the tensor product of two regular graphs to admit Laplacian perfect pair state transfer, where one of the two regular graphs admits perfect state transfer or Laplacian perfect pair state transfer. Additionally, we characterize the existence of Laplacian perfect pair state transfer in the double cover of two regular graphs. By our results, a variety of families of graphs admitting Laplacian perfect pair state transfer can be constructed.
{"title":"Pair state transfer in tensor product and double cover","authors":"Ming Jiang , Xiaogang Liu , Jing Wang","doi":"10.1016/j.dam.2025.12.053","DOIUrl":"10.1016/j.dam.2025.12.053","url":null,"abstract":"<div><div>Quantum state transfer, first introduced by Bose in 2003, is an important physical phenomenon in quantum networks, which plays a vital role in quantum communication and quantum computing. In 2004, Christandl et al. proposed the concept of perfect state transfer on graphs by modeling the quantum network using graphs, and unveiled the feasibility of applying graph theory to quantum state transfer. In 2018, Chen and Godsil proposed the definition of Laplacian perfect pair state transfer on graphs, which is a brilliant generalization of perfect state transfer. In this paper, we investigate the existence of Laplacian perfect pair state transfer in tensor product and double cover of two regular graphs, respectively, and reveal fundamental connections between perfect state transfer and Laplacian perfect pair state transfer. We give necessary and sufficient conditions for the tensor product of two regular graphs to admit Laplacian perfect pair state transfer, where one of the two regular graphs admits perfect state transfer or Laplacian perfect pair state transfer. Additionally, we characterize the existence of Laplacian perfect pair state transfer in the double cover of two regular graphs. By our results, a variety of families of graphs admitting Laplacian perfect pair state transfer can be constructed.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 165-176"},"PeriodicalIF":1.0,"publicationDate":"2026-01-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145928983","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 : 2026-01-05DOI: 10.1016/j.dam.2025.12.049
Yifan Li , Shuming Zhou , Qifan Zhang
For a network , the subversion at the vertex set (resp., edge set) of is defined as the removal of the closed neighborhood of the vertex set (resp., all end vertices of the edge set) from , where the vertex set (resp., edge set) is referred as subverted vertices (resp., edges). Neighbor connectivity and edge neighbor connectivity serve as key indicators for assessing the subversion of spy networks and network disruptions throughout the deletion of closed neighborhood. The neighbor connectivity (resp., edge neighbor connectivity ) of a network is defined as the minimum number of subverted vertices (resp., edges) required to disconnect it, make it empty or complete (resp., trivial). Gu et al. (IEEE Trans. Netw. Sci. Eng. 11 (5) (2024) 1-13) conjectured that whether holds for all compound graphs constructed by the underlying block . In this paper, we solve this conjecture and determine the (edge) neighbor connectivity of a class of hypercube-based compound network, including half hypercube, hierarchical hypercube, hierarchical cubic network and dual-cube-like network. In addition, we present network vulnerability analysis algorithms based on neighborhood fault pattern. To evaluate their effectiveness, taking the half hypercube, hierarchical cubic network and real-world network dwt-918 as examples, we perform experimental simulations to analyze both the cardinality distribution of subverted vertices and topological configurations of survival graph.
对于网络G，在顶点集(p。，边集)定义为顶点集（resp.）的闭邻域的移除。，边集的所有端点)来自G，其中顶点集(resp。，边集)被称为颠覆顶点(如。,边缘)。邻居连通性和边缘邻居连通性是评估间谍网络颠覆和网络中断的关键指标。邻居连通性κNB(G)。，网络G的边缘邻居连通性λNB(G)定义为颠覆顶点的最小个数(p。需要断开它，使其为空或完整(参见。琐碎的)。Gu et al. （IEEE译）Netw。科学。Eng. 11（5)(2024) 1-13）推测κNB(G)=δ(G)−12+1是否对所有由底层块Qn构造的复合图G成立。本文解决了这一猜想，并确定了一类基于超立方体的复合网络（包括半超立方体、分层超立方体、分层立方网络和双类立方体网络）的（边）邻居连通性。此外，提出了基于邻域故障模式的网络漏洞分析算法。为了评估它们的有效性，我们以半超立方体、分层立方网络和现实世界网络dwt-918为例，进行了实验模拟，分析了颠覆顶点的基数分布和生存图的拓扑构型。
{"title":"Neighbor connectivity of hypercube-based compound network","authors":"Yifan Li , Shuming Zhou , Qifan Zhang","doi":"10.1016/j.dam.2025.12.049","DOIUrl":"10.1016/j.dam.2025.12.049","url":null,"abstract":"<div><div>For a network <span><math><mi>G</mi></math></span>, the subversion at the vertex set (resp., edge set) of <span><math><mi>G</mi></math></span> is defined as the removal of the closed neighborhood of the vertex set (resp., all end vertices of the edge set) from <span><math><mi>G</mi></math></span>, where the vertex set (resp., edge set) is referred as subverted vertices (resp., edges). Neighbor connectivity and edge neighbor connectivity serve as key indicators for assessing the subversion of spy networks and network disruptions throughout the deletion of closed neighborhood. The neighbor connectivity <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>N</mi><mi>B</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> (resp., edge neighbor connectivity <span><math><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mi>N</mi><mi>B</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span>) of a network <span><math><mi>G</mi></math></span> is defined as the minimum number of subverted vertices (resp., edges) required to disconnect it, make it empty or complete (resp., trivial). Gu et al. (IEEE Trans. Netw. Sci. Eng. 11 (5) (2024) 1-13) conjectured that whether <span><math><mrow><msub><mrow><mi>κ</mi></mrow><mrow><mi>NB</mi></mrow></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mfenced><mrow><mfrac><mrow><mi>δ</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn></mrow></math></span> holds for all compound graphs <span><math><mi>G</mi></math></span> constructed by the underlying block <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. In this paper, we solve this conjecture and determine the (edge) neighbor connectivity of a class of hypercube-based compound network, including half hypercube, hierarchical hypercube, hierarchical cubic network and dual-cube-like network. In addition, we present network vulnerability analysis algorithms based on neighborhood fault pattern. To evaluate their effectiveness, taking the half hypercube, hierarchical cubic network and real-world network dwt-918 as examples, we perform experimental simulations to analyze both the cardinality distribution of subverted vertices and topological configurations of survival graph.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"384 ","pages":"Pages 1-15"},"PeriodicalIF":1.0,"publicationDate":"2026-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145898206","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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