{"title":"A note proving the nullity of block graphs is unbounded","authors":"","doi":"10.1016/j.disc.2024.114289","DOIUrl":"10.1016/j.disc.2024.114289","url":null,"abstract":"<div><div>Block graphs are important baseline structures for a vast array of community detection and other network partitioning models. Singular graphs have many important uses in the physical sciences. A recent conjecture was posited that the nullity of a <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free block graph cannot be larger than one. In this paper we prove that the conjecture is false by constructing a family of counterexamples using the Cauchy interlacing theorem for real symmetric matrices. In doing so, we prove the stronger statement that the nullity of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>-free block graphs is unbounded. Finally, the implications of this result for the computational network theory literature are discussed.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142445873","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":"Bipartite Ramsey number pairs that involve combinations of cycles and odd paths","authors":"","doi":"10.1016/j.disc.2024.114283","DOIUrl":"10.1016/j.disc.2024.114283","url":null,"abstract":"<div><div>For bipartite graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, the bipartite Ramsey number <span><math><mi>b</mi><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, <span><math><mo>…</mo><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>)</mo></math></span> is the least positive integer <em>b</em>, so that any coloring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>b</mi><mo>,</mo><mi>b</mi></mrow></msub></math></span> with <em>k</em> colors, will result in a copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> in the <em>i</em>th color, for some <em>i</em>. For bipartite graphs <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span>, the bipartite Ramsey number pair <span><math><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, denoted by <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo>)</mo></math></span>, is an ordered pair of integers such that for any blue-red coloring of the edges of <span><math><msub><mrow><mi>K</mi></mrow><mrow><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>,</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span>, with <span><math><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span>, either a blue copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> exists or a red copy of <span><math><msub><mrow><mi>G</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> exists if and only if <span><math><msup><mrow><mi>a</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><mi>a</mi></math></span> and <span><math><msup><mrow><mi>b</mi></mrow><mrow><mo>′</mo></mrow></msup><mo>≥</mo><mi>b</mi></math></span>. In <span><span>[4]</span></span>, Faudree and Schelp considered bipartite Ramsey number pairs involving paths. Recently, Joubert, Hattingh and Henning showed, in <span><span>[7]</span></span> and <span><span>[8]</span></span>, that <span><math><msub><mrow><mi>b</mi></mrow><mrow><mi>p</mi></mrow></msub><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>s</mi></mrow></msub><mo>)</mo><mo>=</mo><mo>(</mo><mn>2</mn><mi>s</mi><mo>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142441529","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":"A method for constructing graphs with the same resistance spectrum","authors":"","doi":"10.1016/j.disc.2024.114284","DOIUrl":"10.1016/j.disc.2024.114284","url":null,"abstract":"<div><div>Let <em>G</em> be a simple graph with vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> and edge set <span><math><mi>E</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The resistance distance <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> between two vertices <span><math><mi>x</mi><mo>,</mo><mi>y</mi></math></span> of <em>G</em>, is defined to be the effective resistance between the two vertices in the corresponding electrical network in which each edge of <em>G</em> is replaced by a unit resistor. The resistance spectrum <span><math><mi>RS</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> is the multiset of the resistance distances between all pairs of vertices in the graph. This paper presents a novel method for constructing graphs with the same resistance spectrum. It is obtained that for any positive integer <em>k</em>, there exist at least <span><math><msup><mrow><mn>2</mn></mrow><mrow><mi>k</mi></mrow></msup></math></span> graphs with the same resistance spectrum. Furthermore, it is shown that for <span><math><mi>n</mi><mo>≥</mo><mn>10</mn></math></span>, there are at least <span><math><mn>2</mn><mo>(</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>10</mn><mo>)</mo><mi>p</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo><mo>+</mo><mi>q</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo><mo>)</mo></math></span> pairs of graphs of order <em>n</em> with the same resistance spectrum, where <span><math><mi>p</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo></math></span> and <span><math><mi>q</mi><mo>(</mo><mi>n</mi><mo>−</mo><mn>9</mn><mo>)</mo></math></span> are the numbers of partitions of the integer <span><math><mi>n</mi><mo>−</mo><mn>9</mn></math></span> and simple graphs of order <span><math><mi>n</mi><mo>−</mo><mn>9</mn></math></span>, respectively.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142434201","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":"Star-critical Ramsey numbers involving large books","authors":"","doi":"10.1016/j.disc.2024.114270","DOIUrl":"10.1016/j.disc.2024.114270","url":null,"abstract":"<div><div>For graphs <span><math><mi>F</mi><mo>,</mo><mi>G</mi></math></span> and <em>H</em>, let <span><math><mi>F</mi><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> signify that any red/blue edge coloring of <em>F</em> contains either a red <em>G</em> or a blue <em>H</em>. The Ramsey number <span><math><mi>r</mi><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span> is defined to be the smallest integer <em>r</em> such that <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>r</mi></mrow></msub><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><mi>H</mi><mo>)</mo></math></span>. Let <span><math><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup></math></span> be the book graph which consists of <em>n</em> copies of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span> all sharing a common <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, and let <span><math><mi>G</mi><mo>:</mo><mo>=</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>(</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>p</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>)</mo></math></span> be the complete <span><math><mo>(</mo><mi>p</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-partite graph with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span>, <span><math><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span> and <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></math></span>.</div><div>In this paper, avoiding the use of Szemerédi's regularity lemma, we show that for any fixed <span><math><mi>p</mi><mo>≥</mo><mn>1</mn></math></span>, <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> and sufficiently large <em>n</em>, <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>p</mi><mo>(</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mi>k</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>+</mo><mn>1</mn></mrow></msub><mo>∖</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn><mo>,</mo><mi>n</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>−</mo><mn>2</mn></mrow></msub><mo>→</mo><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></msubsup><mo>)</mo></math></span>. This implies that the star-critical Ramsey number <span><math><msub><mrow><mi>r</mi></mrow><mrow><mo>⁎</mo></mrow></msub><mo>(</mo><mi>G</mi><mo>,</mo><msubsup><mrow><mi>B</mi></mrow><m","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426770","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 Mostar index of Tribonacci cubes","authors":"","doi":"10.1016/j.disc.2024.114281","DOIUrl":"10.1016/j.disc.2024.114281","url":null,"abstract":"<div><div>Tribonacci cubes <span><math><msubsup><mrow><mi>Γ</mi></mrow><mrow><mi>n</mi></mrow><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></msubsup></math></span> are a class of hypercube-like cubes obtained by removing all vertices of hypercubes <span><math><msub><mrow><mi>Q</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> that have more than two consecutive 1s. In this paper, we calculate the Mostar index of Tribonacci cubes, which is a measure of how far the graph is from being distance-balanced and is used to study various properties of chemical graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426771","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":"Structural parameters of Schnyder woods","authors":"","doi":"10.1016/j.disc.2024.114282","DOIUrl":"10.1016/j.disc.2024.114282","url":null,"abstract":"<div><div>We study two fundamental parameters of Schnyder woods by exploiting structurally related methods. First, we prove a new lower bound on the total number of leaves in the three trees of a Schnyder wood. Second, it is well-known that Schnyder woods can be used to find three compatible ordered path partitions. We prove new lower bounds on the number of singletons, i.e. paths that consists of exactly one vertex, in such compatible ordered path partitions. All bounds that we present are tight.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426769","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":"On isometry and equivalence of skew constacyclic codes","authors":"","doi":"10.1016/j.disc.2024.114279","DOIUrl":"10.1016/j.disc.2024.114279","url":null,"abstract":"<div><div>In this paper we generalize the notion of <em>n</em>-isometry and <em>n</em>-equivalence relation introduced by Chen et al. in <span><span>[13]</span></span>, <span><span>[12]</span></span> to classify constacyclic codes of length <em>n</em> over a finite field <span><math><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow></mrow><mrow><mi>q</mi></mrow></msub></mrow></msub></math></span>, where <span><math><mi>q</mi><mo>=</mo><msup><mrow><mi>p</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> is a prime power, to the case of skew constacyclic codes without derivation. We call these relations respectively <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-equivalence and <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-isometric relation, where <em>n</em> is the length of the code and <em>σ</em> is an automorphism of the finite field. We compute the number of <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-equivalence and <span><math><mo>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo>)</mo></math></span>-isometric classes, and we give conditions on <em>λ</em> and <em>μ</em> for which <span><math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></math></span>-constacyclic codes and <span><math><mo>(</mo><mi>σ</mi><mo>,</mo><mi>μ</mi><mo>)</mo></math></span>-constacyclic codes are equivalent. Under some conditions on <em>n</em> and <em>q</em> we prove that skew constacyclic codes are equivalent to cyclic codes by using properties of our equivalence relation introduced. We also prove that when <em>q</em> is even and <em>σ</em> is the Frobenius automorphism, skew constacyclic codes of length <em>n</em> are equivalent to cyclic codes when <span><math><mi>gcd</mi><mo></mo><mo>(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo>)</mo><mo>=</mo><mn>1</mn></math></span>. Finally we give some examples as applications of the theory developed here.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416804","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 cyclic symmetric Hamilton cycle decompositions of complete multipartite graphs","authors":"","doi":"10.1016/j.disc.2024.114277","DOIUrl":"10.1016/j.disc.2024.114277","url":null,"abstract":"<div><div>A decomposition of a graph with <em>n</em> vertices, labeled by <span><math><msub><mrow><mi>Z</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, is cyclic if addition by 1 to the vertices acts on the decomposition, and the decomposition is <em>d</em>-symmetric for a divisor <em>d</em> of <em>n</em> if addition by <span><math><mi>n</mi><mo>/</mo><mi>d</mi></math></span> to the vertices acts invariantly on the decomposition. In a 2017 paper, Merola et al. established the necessary and sufficient conditions under which a complete multipartite graph with an even number of parts, each with <em>d</em> vertices, has a cyclic Hamilton cycle decomposition; these decompositions were also <em>d</em>-symmetric.</div><div>In this paper we establish the necessary and sufficient conditions for the analogous question with complete multipartite graphs with an odd number of parts, which settles the existence of cyclic, <em>d</em>-symmetric Hamilton cycle decompositions for all balanced, complete multipartite graphs.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416657","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":"Octopuses in the Boolean cube: Families with pairwise small intersections, part II","authors":"","doi":"10.1016/j.disc.2024.114280","DOIUrl":"10.1016/j.disc.2024.114280","url":null,"abstract":"<div><div>The problem we consider originally arises from 2-level polytope theory. This class of polytopes generalizes a number of other polytope families. One of the important questions in this field can be formulated as follows: is it true for a <em>d</em>-dimensional 2-level polytope that the product of the number of its vertices and the number of its <span><math><mi>d</mi><mo>−</mo><mn>1</mn></math></span> dimensional facets is bounded by <span><math><mi>d</mi><msup><mrow><mn>2</mn></mrow><mrow><mi>d</mi><mo>−</mo><mn>1</mn></mrow></msup></math></span>? Recently, Kupavskii and Weltge <span><span>[9]</span></span> settled this question in positive. A key element in their proof is a more general result for families of vectors in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>d</mi></mrow></msup></math></span> such that the scalar product between any two vectors from different families is either 0 or 1.</div><div>Peter Frankl noted that, when restricted to the Boolean cube, the solution boils down to an elegant application of the Harris–Kleitman correlation inequality. Meanwhile, this problem becomes much more sophisticated when we consider several families.</div><div>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> be families of subsets of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. We suppose that for distinct <span><math><mi>k</mi><mo>,</mo><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></math></span> and arbitrary <span><math><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>∈</mo><msub><mrow><mi>F</mi></mrow><mrow><msup><mrow><mi>k</mi></mrow><mrow><mo>′</mo></mrow></msup></mrow></msub></math></span> we have <span><math><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>∩</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>⩽</mo><mi>m</mi></math></span>. We are interested in the maximal value of <span><math><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>…</mo><mo>|</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>|</mo></math></span> and the structure of the extremal example.</div><div>In the previous paper on the topic, the authors found the asymptotics of this product for constant <em>ℓ</em> and <em>m</em> as <em>n</em> tends to infinity. However, the possible structure of the families from the extremal example turned out to be very complicated. In this paper, we obtain a strong structural result for the extremal families.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142426772","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 proper interval completion problem within some chordal subclasses","authors":"","doi":"10.1016/j.disc.2024.114274","DOIUrl":"10.1016/j.disc.2024.114274","url":null,"abstract":"<div><div>Given a property (graph class) Π, a graph <em>G</em>, and an integer <em>k</em>, the Π<em>-completion</em> problem consists of deciding whether we can turn <em>G</em> into a graph with the property Π by adding at most <em>k</em> edges to <em>G</em>. The Π-completion problem is known to be NP-hard for general graphs when Π is the property of being a proper interval graph (PIG). In this work, we study the PIG-completion problem within different subclasses of chordal graphs. We show that the problem remains NP-complete even when restricted to split graphs. We then turn our attention to positive results and present polynomial time algorithms to solve the PIG-completion problem when the input is restricted to caterpillar and threshold graphs. We also present an efficient algorithm for the minimum co-bipartite-completion for quasi-threshold graphs, which provides a lower bound for the PIG-completion problem within this graph class.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-10-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142416656","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}
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