Discrete Applied Mathematics最新文献

{"title":"Nut digraphs","authors":"Nino Bašić ,&nbsp;Patrick W. Fowler ,&nbsp;Maxine M. McCarthy ,&nbsp;Primož Potočnik","doi":"10.1016/j.dam.2025.12.037","DOIUrl":"10.1016/j.dam.2025.12.037","url":null,"abstract":"<div><div>A <em>nut graph</em> is a simple graph whose kernel is spanned by a single full vector (i.e., the adjacency matrix has a single zero eigenvalue and all non-zero kernel eigenvectors have no zero entry). We classify generalisations of nut graphs to nut digraphs: a digraph whose kernel (resp. co-kernel) is spanned by a full vector is <em>dextro-nut</em> (resp. <em>laevo-nut</em>); a <em>bi-nut</em> digraph is both laevo- and dextro-nut; an <em>ambi-nut</em> digraph is a bi-nut digraph where kernel and co-kernel are spanned by the same vector; a digraph is <em>inter-nut</em> if the intersection of the kernel and co-kernel is spanned by a full vector. It is known that a nut graph is connected, leafless and non-bipartite. It is shown here that an ambi-nut digraph is strongly connected, non-bipartite (i.e., has a non-bipartite underlying graph) and has minimum in-degree and minimum out-degree of at least 2. Refined notions of core and core-forbidden vertices apply to singular digraphs. Infinite families of nut digraphs and systematic coalescence, crossover and multiplier constructions are introduced. Relevance of nut digraphs to topological physics is discussed.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 203-226"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842074","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 hypergraph orientation problem with vertex constraints","authors":"Alberto José Ferrari ,&nbsp;Valeria Leoni ,&nbsp;Graciela Nasini ,&nbsp;Gabriel Valiente","doi":"10.1016/j.dam.2026.01.003","DOIUrl":"10.1016/j.dam.2026.01.003","url":null,"abstract":"<div><div>In computational biology and bioinformatics, hypergraphs model metabolic pathways and networks representing compounds as vertices and reactions as hyperedges. In a previous work we considered the problem of assigning a direction to the hyperedges of a hypergraph minimizing the number of source and sink vertices. We proved that this problem is NP-hard and that it is polynomial-time solvable on graphs.</div><div>In a more general setting, a compound can be a source or a sink in a particular metabolic pathway but, in the context of a metabolic network, it may become both a sink of one pathway and a source of another pathway (an internal vertex). Therefore, in the present work we address a more general form of the hypergraph orientation problem in which some vertices are constrained to be a source, a sink, or an internal vertex. We prove that it remains polynomial-time solvable on graphs by giving a linear-time algorithm. We propose a polynomial-size ILP formulation of the problem, which, applied to the biochemical reactions stored in the Kyoto Encyclopedia of Genes and Genomes (KEGG) database, shows that metabolic pathways and networks, and random hypergraphs with thousands of vertices and hyperedges, can be oriented in a few seconds on a personal computer.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 355-366"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145977709","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":"New perspectives on semiring applications to dynamic programming","authors":"Ambroise Baril ,&nbsp;Miguel Couceiro ,&nbsp;Victor Lagerkvist","doi":"10.1016/j.dam.2025.12.035","DOIUrl":"10.1016/j.dam.2025.12.035","url":null,"abstract":"<div><div>Semiring algebras have been shown to provide a suitable language to formalize many noteworthy combinatorial problems. For instance, the <span>Shortest-Path</span> problem can be seen as a special case of the <span>Algebraic-Path</span> problem when applied to the tropical semiring. The application of semirings typically makes it possible to solve extended problems without increasing the computational complexity. In this article we further exploit the idea of using semiring algebras to address and tackle several extensions of classical computational problems by dynamic programming.</div><div>We consider a general approach which allows us to define a semiring extension of <em>any</em> problem with a reasonable notion of a certificate (e.g., an <span>NP</span> problem). This allows us to consider cost variants of these combinatorial problems, as well as their counting extensions where the goal is to determine how many solutions a given problem admits. The approach makes no particular assumptions (such as idempotence) on the semiring structure. We also propose a new associative algebraic operation on semirings, called <span><math><mi>Δ</mi></math></span>-product, which enables our dynamic programming algorithms to count the number of solutions of minimal costs. We illustrate the advantages of our framework on two well-known but computationally very different <span>NP</span>-hard problems, namely, <span>Connected-Dominating-Set</span> problems and finite-domain <span>Constraint Satisfaction Problems</span> (<span>Csp</span>s). In particular, we prove fixed parameter tractability (<span>FPT</span>) with respect to clique-width and tree-width of the input. This also allows us to count solutions of minimal cost, which is an overlooked problem in the literature.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 243-279"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842076","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":"Maximal non-empty cross s-union families","authors":"Yongjiang Wu ,&nbsp;Yongtao Li ,&nbsp;Zhiyi Liu ,&nbsp;Lihua Feng ,&nbsp;Tingzeng Wu","doi":"10.1016/j.dam.2025.12.038","DOIUrl":"10.1016/j.dam.2025.12.038","url":null,"abstract":"<div><div>Two families of sets <span><math><mi>F</mi></math></span> and <span><math><mi>G</mi></math></span> are said to be cross <span><math><mi>s</mi></math></span>-union if for any <span><math><mrow><mi>F</mi><mo>∈</mo><mi>F</mi></mrow></math></span> and <span><math><mrow><mi>G</mi><mo>∈</mo><mi>G</mi></mrow></math></span>, <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>∪</mo><mi>G</mi><mo>|</mo></mrow><mo>≤</mo><mi>s</mi></mrow></math></span>. In 2021, Frankl and Wong proved that if <span><math><mrow><mi>F</mi><mo>,</mo><mi>G</mi><mo>⊆</mo><msup><mrow><mn>2</mn></mrow><mrow><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow></mrow></msup></mrow></math></span> are non-empty cross <span><math><mi>s</mi></math></span>-union, then <span><math><mrow><mrow><mo>|</mo><mi>F</mi><mo>|</mo></mrow><mo>+</mo><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>≤</mo><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>s</mi></mrow></msubsup><mfenced><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mi>i</mi></mrow></mfrac></mrow></mfenced><mo>+</mo><mn>1</mn><mo>.</mo></mrow></math></span> Moreover, for <span><math><mrow><mi>s</mi><mo>&lt;</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>, equality holds if and only if <span><math><mrow><mfenced><mrow><mi>F</mi><mo>,</mo><mi>G</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mrow><mo>{</mo><mo>0̸</mo><mo>}</mo></mrow><mo>,</mo><mrow><mo>{</mo><mi>G</mi><mo>⊆</mo><mrow><mo>[</mo><mi>n</mi><mo>]</mo></mrow><mo>:</mo><mrow><mo>|</mo><mi>G</mi><mo>|</mo></mrow><mo>≤</mo><mi>s</mi><mo>}</mo></mrow></mrow></mfenced></mrow></math></span>. In this paper, we give a new method to prove this result. Our method also allows us to establish a vector space version and a hereditary family extension. As a byproduct, we revisit the vector space version of the Katona <span><math><mi>s</mi></math></span>-union theorem due to Frankl and Tokushige, and characterize the extremal families for the case <span><math><mrow><mi>s</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 130-138"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792158","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 upper bound on path cover number of regular graphs and its application to Hamiltonian cycle in tough graphs","authors":"Xiaodan Chen,&nbsp;Xiaoning Yang","doi":"10.1016/j.dam.2025.12.036","DOIUrl":"10.1016/j.dam.2025.12.036","url":null,"abstract":"<div><div>The path cover number of a graph <span><math><mi>G</mi></math></span> is the minimum integer <span><math><mi>β</mi></math></span> such that <span><math><mi>G</mi></math></span> contains <span><math><mi>β</mi></math></span> vertex-disjoint paths that cover all of its vertices. In this paper, we first establish an upper bound on the path cover number for regular graphs. Then we apply this bound to help to derive sufficient conditions for a <span><math><mi>t</mi></math></span>-tough graph to be Hamiltonian with integer <span><math><mrow><mi>t</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, in terms of the edge number of the graph, which improve some known results in the literature. Another key tool we used to derive these sufficient conditions is the (complete) toughness closure lemma due to Hoàng and Robin (2024) and Shan and Tanyel (2025).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 113-121"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792205","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":"Proof of a conjecture on graph polytope","authors":"Feihu Liu","doi":"10.1016/j.dam.2025.12.034","DOIUrl":"10.1016/j.dam.2025.12.034","url":null,"abstract":"<div><div>Graph polytopes arising from vertex-weighted graphs were first introduced by Bóna, Ju, and Yoshida. We prove a conjecture stating that for any simple connected graph, the numerator polynomial of the Ehrhart series of its graph polytope is palindromic, using Stanley’s reciprocity theorem. Furthermore, we introduce hypergraph polytopes and establish that every simple, connected, unimodular hypergraph polytope is an integer polytope. Additionally, for simple connected uniform hypergraph polytopes, we demonstrate that the numerator polynomial of their Ehrhart series is palindromic.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 122-129"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792199","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":"Semidefinite programming bounds and a Branch-and-bound algorithm for the Chordless Cycle Problem","authors":"Dilson Lucas Pereira ,&nbsp;Dilson Almeida Guimarães ,&nbsp;Alexandre Salles da Cunha ,&nbsp;Abilio Lucena","doi":"10.1016/j.dam.2026.01.015","DOIUrl":"10.1016/j.dam.2026.01.015","url":null,"abstract":"<div><div>In this paper, we investigate the Chordless Cycle Problem (CCP), that asks for a maximum cardinality set of vertices whose induced subgraph is a cycle for a given connected undirected graph. In order to solve the CCP exactly, we propose an enhanced Lagrangian Relaxation Algorithm (LRA) and a Branch-and-bound algorithm, <span>BBLAGSDP</span>, that relies on the LRA. Enhancements come mostly from the fact that the matrix <span><math><mi>Λ</mi></math></span> of Lagrangian multipliers attached to the semidefinite programming (SDP) constraint involved in our relaxation for the CCP is not only positive semidefinite, but also symmetric. For this reason, we are allowed to replace <span><math><mi>Λ</mi></math></span> by its factorization <span><math><mrow><mi>Λ</mi><mo>=</mo><mi>Γ</mi><msup><mrow><mi>Γ</mi></mrow><mrow><mi>T</mi></mrow></msup></mrow></math></span>, reformulate its accompanying Lagrangian Dual Problem (LDP) in terms of <span><math><mi>Γ</mi></math></span> and, finally, avoid the need for computing an eigendecomposition of <span><math><mi>Λ</mi></math></span>, that would otherwise be the most expensive step required for solving the LDP. On the one hand, our LRA approximates very well the exact SDP bounds, computed by a convex optimization solver from the literature. And it does so in significantly smaller computational times. On the other hand, our SDP bounds are also much stronger than the previously available Linear Programming (LP) bounds. Computational experiments conduced with 458 instances indicate that <span>BBLAGSDP</span> is by far the best performing algorithm, among the five exact methods we compare here, as the density of the input graph increases.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 387-408"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"146023130","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 Kőnig–Egerváry index of a graph","authors":"Daniel A. Jaume ,&nbsp;Vadim E. Levit ,&nbsp;Eugen Mandrescu ,&nbsp;Gonzalo Molina ,&nbsp;Kevin Pereyra","doi":"10.1016/j.dam.2025.12.039","DOIUrl":"10.1016/j.dam.2025.12.039","url":null,"abstract":"<div><div>A graph is said to be Kőnig–Egerváry if its matching number equals its vertex cover number. The difference between these two graph parameters, the vertex cover number minus the matching number, measures, in some sense, how far a graph is from being a Kőnig–Egerváry graph. Several properties of this difference, called the Kőnig–Egerváry index or Kőnig deficiency, are presented, including some nontrivial structural characterizations. Furthermore, it is shown that various statements involving Kőnig–Egerváry graphs are, in fact, general statements about graphs that can be expressed in terms of their Kőnig–Egerváry indices.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 139-151"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145792159","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":"Factors of bipartite graphs with degree conditions imposed on each partite set","authors":"Michitaka Furuya ,&nbsp;Mikio Kano","doi":"10.1016/j.dam.2025.12.027","DOIUrl":"10.1016/j.dam.2025.12.027","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a bipartite graph with bipartition <span><math><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo>)</mo></mrow></math></span>, and let <span><math><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>:</mo><mi>X</mi><mo>→</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mi>c</mi><mo>:</mo><mi>Y</mi><mo>→</mo><msub><mrow><mi>Z</mi></mrow><mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow></math></span> be functions. In this paper, we give a sufficient condition for <span><math><mi>G</mi></math></span> to have a factor <span><math><mi>F</mi></math></span> satisfying <span><math><mrow><msub><mrow><mo>deg</mo></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mi>a</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>,</mo><mi>b</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>}</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow></math></span> and <span><math><mrow><msub><mrow><mo>deg</mo></mrow><mrow><mi>F</mi></mrow></msub><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>∈</mo><mrow><mo>{</mo><mi>c</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>,</mo><mi>c</mi><mrow><mo>(</mo><mi>y</mi><mo>)</mo></mrow><mo>+</mo><mn>1</mn><mo>}</mo></mrow></mrow></math></span> for all <span><math><mrow><mi>y</mi><mo>∈</mo><mi>Y</mi></mrow></math></span>. Our theorem modifies a result in Addario-Berry et al. (2008).</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 165-168"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145842055","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":"Spectral extrema of graphs: Forbidden linear forests and non-bipartite graphs","authors":"Xuelian Mao,&nbsp;Zhenyu Ni,&nbsp;Ming-Zhu Chen","doi":"10.1016/j.dam.2025.12.017","DOIUrl":"10.1016/j.dam.2025.12.017","url":null,"abstract":"<div><div>For a set of graphs <span><math><mi>F</mi></math></span>, a graph <span><math><mi>G</mi></math></span> is called <span><math><mi>F</mi></math></span>-free if it does not contain any graph in <span><math><mi>F</mi></math></span> as a subgraph. Let SPEX<span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mi>F</mi><mo>)</mo></mrow></math></span> denote the graphs with the maximum spectral radius among all <span><math><mi>F</mi></math></span>-free graphs of order <span><math><mi>n</mi></math></span>. In this paper, for any non-bipartite graph <span><math><mi>H</mi></math></span>, we give some characterizations for the graphs in SPEX<span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>ℓ</mi></mrow></msubsup><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>,</mo><mi>H</mi><mo>}</mo></mrow><mo>)</mo></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span>, where <span><math><mrow><mi>ℓ</mi><mo>≥</mo><mn>1</mn></mrow></math></span>, <span><math><mrow><msub><mrow><mi>k</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mo>⋯</mo><mo>≥</mo><msub><mrow><mi>k</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>≥</mo><mn>2</mn></mrow></math></span>, and there exists at least one <span><math><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> not equal to 3. As an application, we completely characterize the graphs in SPEX<span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>ℓ</mi></mrow></msubsup><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></math></span> and SPEX<span><math><mrow><mo>(</mo><mi>n</mi><mo>,</mo><mrow><mo>{</mo><msubsup><mrow><mo>⋃</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>ℓ</mi></mrow></msubsup><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>}</mo></mrow><mo>)</mo></mrow></math></span> for sufficiently large <span><math><mi>n</mi></math></span>, where <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a friendship graph on <span><math><mrow><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></mrow></math></span> vertices consisting of <span><math><mi>k</mi></math></span> triangles which intersect in exactly one common vertex.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"383 ","pages":"Pages 8-14"},"PeriodicalIF":1.0,"publicationDate":"2026-04-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145748178","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}