{"title":"Longest subsequence for certain repeated up/down patterns in random permutations avoiding a pattern of length three","authors":"Ross G. Pinsky","doi":"10.1016/j.disc.2025.114784","DOIUrl":"10.1016/j.disc.2025.114784","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> denote the set of permutations of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span> and let <span><math><mi>σ</mi><mo>=</mo><msub><mrow><mi>σ</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. For any subsequence <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo>}</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>k</mi></mrow></msubsup></math></span> of <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>σ</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span> of length <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span>, construct the “up/down” sequence <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub></math></span> defined by<span><span><span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>=</mo><mrow><mo>{</mo><mtable><mtr><mtd><mi>U</mi><mo>,</mo><mspace></mspace><mtext>if</mtext><mspace></mspace><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>−</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo>></mo><mn>0</mn><mo>;</mo></mtd></mtr><mtr><mtd><mi>D</mi><mo>,</mo><mspace></mspace><mtext>if</mtext><mspace></mspace><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>−</mo><msub><mrow><mi>σ</mi></mrow><mrow><msub><mrow><mi>i</mi></mrow><mrow><mi>j</mi></mrow></msub></mrow></msub><mo><</mo><mn>0</mn><mo>,</mo></mtd></mtr></mtable></mrow></math></span></span></span> where <em>U</em> refers to “up”, <em>D</em> to “down” and <em>V</em> to “vertical”. Consider now a fixed up/down pattern: <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>l</mi></mrow></msub></math></span>, where <span><math><mi>l</mi><mo>∈</mo><mi>N</mi></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>∈</mo><mo>{</mo><mi>U</mi><mo>,</mo><mi>D</mi><mo>}</mo><mo>,</mo><mspace></mspace><mi>j</mi><mo>∈</mo><mo>[</mo><mi>l</mi><mo>]</mo></math></span>. Given a permutation <span><math><mi>σ</mi><mo>∈</mo><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>, consider the length of the longest subsequence of <em>σ</em> that repeats this pattern. Incomplete patterns are not counted, so the length is necessarily either 0 or of the form <span><mat","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114784"},"PeriodicalIF":0.7,"publicationDate":"2025-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018505","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":"Vertex connectivity of chordal graphs","authors":"Tài Huy Hà , Takayuki Hibi","doi":"10.1016/j.disc.2025.114777","DOIUrl":"10.1016/j.disc.2025.114777","url":null,"abstract":"<div><div>Let <em>G</em> be a finite graph and <span><math><mi>ϰ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> the vertex connectivity of <em>G</em>. A chordal graph <em>G</em> is called chordal<sup>⁎</sup> if no vertex of <em>G</em> is adjacent to all other vertices of <em>G</em>. Using the syzygy theory in commutative algebra, it is proved that every chordal<sup>⁎</sup> graph <em>G</em> on <em>n</em> vertices satisfies <span><math><mi>ϰ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>≤</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>−</mo><mo>⌈</mo><mn>2</mn><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>−</mo><mn>2</mn><mspace></mspace><mo>⌉</mo></math></span>. Furthermore, given an integer <span><math><mn>0</mn><mo>≤</mo><mi>ϰ</mi><mo>≤</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo><mo>−</mo><mo>⌈</mo><mn>2</mn><msqrt><mrow><mi>n</mi></mrow></msqrt><mo>−</mo><mn>2</mn><mspace></mspace><mo>⌉</mo></math></span>, a chordal<sup>⁎</sup> graph <em>G</em> on <em>n</em> vertices satisfying <span><math><mi>ϰ</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>ϰ</mi></math></span> is constructed.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114777"},"PeriodicalIF":0.7,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145019223","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":"Linear Turán problems with bounded matching number in hypergraphs","authors":"Junpeng Zhou , Xiying Yuan","doi":"10.1016/j.disc.2025.114772","DOIUrl":"10.1016/j.disc.2025.114772","url":null,"abstract":"<div><div>An <em>r</em>-uniform hypergraph (<em>r</em>-graph) is linear if any two edges intersect in at most one vertex. Let <span><math><mi>F</mi></math></span> be a given family of <em>r</em>-graphs. A hypergraph <em>H</em> is called <span><math><mi>F</mi></math></span>-free if <em>H</em> does not contain any hypergraphs in <span><math><mi>F</mi></math></span>. The linear Turán number of <span><math><mi>F</mi></math></span> is defined as the maximum number of edges of all <span><math><mi>F</mi></math></span>-free linear <em>r</em>-graphs on <em>n</em> vertices. For a given graph <em>F</em>, the <em>r</em>-expansion of <em>F</em> is the <em>r</em>-graph <span><math><msup><mrow><mi>F</mi></mrow><mrow><mi>r</mi></mrow></msup></math></span> constructed by inserting <span><math><mi>r</mi><mo>−</mo><mn>2</mn></math></span> new distinct vertices in each edge of <em>F</em>.</div><div>Recently, Alon and Frankl (2024) <span><span>[1]</span></span> and Gerbner (2023) <span><span>[10]</span></span> studied the maximum number of edges in <em>F</em>-free graphs on <em>n</em> vertices with bounded matching number, respectively. In this paper, we investigate the analogous linear Turán problems for hypergraphs with bounded matching number. Specifically, the bounds for the linear Turán number of expansions of non-bipartite graphs with bounded matching number are obtained. As an application, the maximum possible number of edges of <span><math><msubsup><mrow><mi>K</mi></mrow><mrow><mi>ℓ</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>r</mi></mrow></msubsup></math></span>-free linear <em>r</em>-graphs with bounded matching number in some cases is determined. Furthermore, the linear Turán number of the expansion of any bipartite graph with bounded matching number is determined.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114772"},"PeriodicalIF":0.7,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010087","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 complexity of power graph associated with finite p-groups","authors":"Sakineh Rahbariyan","doi":"10.1016/j.disc.2025.114778","DOIUrl":"10.1016/j.disc.2025.114778","url":null,"abstract":"<div><div>Given a finite group <em>G</em>, the power graph <span><math><mi>P</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> is the graph with a vertex set consisting of the elements of <em>G</em>, where two vertices <em>x</em> and <em>y</em> are joined by an edge if and only if <span><math><mo>〈</mo><mi>x</mi><mo>〉</mo><mo>⊆</mo><mo>〈</mo><mi>y</mi><mo>〉</mo></math></span> or <span><math><mo>〈</mo><mi>y</mi><mo>〉</mo><mo>⊆</mo><mo>〈</mo><mi>x</mi><mo>〉</mo></math></span>. We consider <span><math><mi>κ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, the number of spanning trees of <span><math><mi>P</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, called the complexity of <span><math><mi>P</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>. The purpose of this paper is twofold. First, we derive some explicit formulas concerning the complexity <span><math><mi>κ</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> for a finite <em>p</em>-group <em>G</em>. Second, we show that the simple group <span><math><msub><mrow><mi>A</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span> is uniquely determined by the number of spanning trees of its power graph among all finite groups.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114778"},"PeriodicalIF":0.7,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145018504","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":"Generalized toughness and spectral radius of graphs","authors":"Yuanyuan Chen , Xiaofeng Gu , Huiqiu Lin","doi":"10.1016/j.disc.2025.114776","DOIUrl":"10.1016/j.disc.2025.114776","url":null,"abstract":"<div><div>The toughness <span><math><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a non-complete graph <em>G</em> is defined as <span><math><mi>t</mi><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo></mo><mo>{</mo><mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></mfrac><mo>}</mo></math></span> in which the minimum is taken over all proper subsets <em>S</em> of <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span> is disconnected, where <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></math></span> denotes the number of components of <span><math><mi>G</mi><mo>−</mo><mi>S</mi></math></span>. For an integer <span><math><mi>l</mi><mo>≥</mo><mn>2</mn></math></span>, we generalize this concept and define the <em>l</em>-toughness <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo></math></span> of a graph <em>G</em> to be <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>=</mo><mi>min</mi><mo></mo><mrow><mo>{</mo><mfrac><mrow><mo>|</mo><mi>S</mi><mo>|</mo></mrow><mrow><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo></mrow></mfrac><mo>}</mo></mrow></math></span>, in which the minimum is taken over all proper subsets <span><math><mi>S</mi><mo>⊂</mo><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> such that <span><math><mi>c</mi><mo>(</mo><mi>G</mi><mo>−</mo><mi>S</mi><mo>)</mo><mo>≥</mo><mi>l</mi></math></span>. If <span><math><msub><mrow><mi>t</mi></mrow><mrow><mi>l</mi></mrow></msub><mo>(</mo><mi>G</mi><mo>)</mo><mo>≥</mo><mi>k</mi></math></span>, then we say that <em>G</em> is <span><math><mo>(</mo><mi>k</mi><mo>,</mo><mi>l</mi><mo>)</mo></math></span>-tough. This generalization of toughness is interesting on its own in graph theory, and we show evidence that it is closely related to generalized connectivity and factors. By incorporating the <em>l</em>-toughness and spectrum, we provide spectral radius conditions for a graph to be <span><math><mo>(</mo><mi>t</mi><mo>,</mo><mi>l</mi><mo>)</mo></math></span>-tough with a positive integer <em>t</em> and to be <span><math><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac><mo>,</mo><mi>l</mi><mo>)</mo></math></span>-tough with an integer <span><math><mi>b</mi><mo>≥</mo><mn>2</mn></math></span>, respectively. Additionally, we also discover a spectral radius condition for <span><math><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mi>b</mi></mrow></mfrac><mo>,</mo><mi>l</mi><mo>)</mo></math></span>-tough graphs with a fixed minimum degree.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114776"},"PeriodicalIF":0.7,"publicationDate":"2025-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145010088","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 determinants of tournaments and the characterization of D5","authors":"Jing Zeng, Lihua You","doi":"10.1016/j.disc.2025.114766","DOIUrl":"10.1016/j.disc.2025.114766","url":null,"abstract":"<div><div>Let <em>T</em> be a tournament with <em>n</em> vertices <span><math><msub><mrow><mi>v</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>v</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span>. The skew-adjacency matrix of <em>T</em> is the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> zero-diagonal matrix <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub><mo>=</mo><mo>[</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>]</mo></math></span> in which <span><math><msub><mrow><mi>s</mi></mrow><mrow><mi>i</mi><mi>j</mi></mrow></msub><mo>=</mo><mo>−</mo><msub><mrow><mi>s</mi></mrow><mrow><mi>j</mi><mi>i</mi></mrow></msub><mo>=</mo><mn>1</mn></math></span> if <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> dominates <span><math><msub><mrow><mi>v</mi></mrow><mrow><mi>j</mi></mrow></msub></math></span>. We define the determinant <span><math><mi>det</mi><mo></mo><mo>(</mo><mi>T</mi><mo>)</mo></math></span> of <em>T</em> as the determinant of <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>T</mi></mrow></msub></math></span>. It is well-known that <span><math><mi>det</mi><mo></mo><mo>(</mo><mi>T</mi><mo>)</mo><mo>=</mo><mn>0</mn></math></span> if <em>n</em> is odd and <span><math><mi>det</mi><mo></mo><mo>(</mo><mi>T</mi><mo>)</mo></math></span> is the square of an odd integer if <em>n</em> is even. Let <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> be the set of tournaments whose all subtournaments have determinant at most <span><math><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, where <em>k</em> is a positive odd integer. The necessary and sufficient condition for <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> or <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mn>3</mn></mrow></msub></math></span> has been characterized in 2023. In this paper, we characterize the set <span><math><msub><mrow><mi>D</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, and we obtain some properties of <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. Moreover, for any positive odd integer <em>k</em>, we give a construction of a tournament <em>T</em> satisfying <span><math><mi>det</mi><mo></mo><mo>(</mo><mi>T</mi><mo>)</mo><mo>=</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>2</mn></mrow></msup></math></span>, and <span><math><mi>T</mi><mo>∈</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>﹨</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msub></math></span> if <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span>, which implies <span><math><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>﹨</mo><msub><mrow><mi>D</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></msub></math></spa","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114766"},"PeriodicalIF":0.7,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996761","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":"Minimum bisections of graphs without even cycles","authors":"Mengjiao Rao , Qinghou Zeng","doi":"10.1016/j.disc.2025.114768","DOIUrl":"10.1016/j.disc.2025.114768","url":null,"abstract":"<div><div>A bisection of a graph <em>G</em> is a partition of <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> into two parts <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> satisfying <span><math><mo>|</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>|</mo><mo>−</mo><mo>|</mo><msub><mrow><mi>V</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>|</mo><mo>|</mo><mo>≤</mo><mn>1</mn></math></span>, and its size is the number of edges that go across the two parts. The minimum bisection problem asks for a bisection in a given graph minimizing the size which is defined as the bisection width. We present some upper bounds on the bisection width for graphs with a perfect matching and without short even cycles. Let <em>G</em> be an <em>n</em>-vertex <span><math><mo>{</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>,</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>}</mo></math></span>-free graph with <em>m</em> edges and a perfect matching. We first show that <em>G</em> has a bisection of size at most <span><math><mi>m</mi><mo>/</mo><mn>2</mn><mo>−</mo><mo>(</mo><mi>n</mi><mo>−</mo><mn>6</mn><mo>)</mo><mo>/</mo><mn>8</mn></math></span>. Together with probabilistic techniques, we show that there is a constant <span><math><mi>ζ</mi><mo>></mo><mn>0</mn></math></span> such that <em>G</em> has bisection of size at most <span><math><mi>m</mi><mo>/</mo><mn>2</mn><mo>−</mo><mi>ζ</mi><mspace></mspace><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><msqrt><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msqrt></math></span> if <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>, and this is tight up to the value of <em>ζ</em>. Furthermore, if <em>G</em> is <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>2</mn><mi>k</mi></mrow></msub></math></span>-free for some fixed integer <span><math><mi>k</mi><mo>≥</mo><mn>3</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>8</mn></math></span>, then there is a constant <span><math><mi>c</mi><mo>(</mo><mi>k</mi><mo>)</mo><mo>></mo><mn>0</mn></math></span> such that <em>G</em> has a bisection of size at most <span><math><mi>m</mi><mo>/</mo><mn>2</mn><mo>−</mo><mi>c</mi><mo>(</mo><mi>k</mi><mo>)</mo><msup><mrow><mi>m</mi></mrow><mrow><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo><mo>/</mo><mo>(</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow></msup></math></span>. This is tight up to the value of <span><math><mi>c</mi><mo>(</mo><mi>k</mi><mo>)</mo></math></span> for <span><math><mn>2</mn><mi>k</mi><mo>∈</mo><mo>{</mo><mn>6</mn><mo>,</mo><mn>10</mn><mo>}</mo></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114768"},"PeriodicalIF":0.7,"publicationDate":"2025-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144996898","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}
Yuchen Ding , Yu-Chen Sun , Li-Yuan Wang , Yutong Xia
{"title":"A note on additive complements of the squares","authors":"Yuchen Ding , Yu-Chen Sun , Li-Yuan Wang , Yutong Xia","doi":"10.1016/j.disc.2025.114763","DOIUrl":"10.1016/j.disc.2025.114763","url":null,"abstract":"<div><div>Let <span><math><mi>S</mi><mo>=</mo><mo>{</mo><msup><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msup><mrow><mn>2</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><msup><mrow><mn>3</mn></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>}</mo></math></span> be the set of squares and <span><math><mi>W</mi><mo>=</mo><msubsup><mrow><mo>{</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mo>∞</mo></mrow></msubsup><mo>⊂</mo><mi>N</mi></math></span> be an additive complement of <span><math><mi>S</mi></math></span> so that <span><math><mi>S</mi><mo>+</mo><mi>W</mi><mo>⊃</mo><mo>{</mo><mi>n</mi><mo>∈</mo><mi>N</mi><mo>:</mo><mi>n</mi><mo>≥</mo><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>}</mo></math></span> for some <span><math><msub><mrow><mi>N</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span>. Let <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>S</mi><mo>,</mo><mi>W</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>=</mo><mi>#</mi><mo>{</mo><mo>(</mo><mi>s</mi><mo>,</mo><mi>w</mi><mo>)</mo><mo>:</mo><mi>n</mi><mo>=</mo><mi>s</mi><mo>+</mo><mi>w</mi><mo>,</mo><mi>s</mi><mo>∈</mo><mi>S</mi><mo>,</mo><mi>w</mi><mo>∈</mo><mi>W</mi><mo>}</mo></math></span>.</div><div>In 2017, Chen-Fang <span><span>[5]</span></span> studied the lower bound of <span><math><msubsup><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></msubsup><msub><mrow><mi>R</mi></mrow><mrow><mi>S</mi><mo>,</mo><mi>W</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. In this note, we improve Cheng-Fang's result and get that<span><span><span><math><munderover><mo>∑</mo><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>N</mi></mrow></munderover><msub><mrow><mi>R</mi></mrow><mrow><mi>S</mi><mo>,</mo><mi>W</mi></mrow></msub><mo>(</mo><mi>n</mi><mo>)</mo><mo>−</mo><mi>N</mi><mo>≫</mo><msup><mrow><mi>N</mi></mrow><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></msup><mo>.</mo></math></span></span></span> We also provide a heuristic argument suggesting that our bound obtained above maybe close to optimal. As an application, we make some progress on a problem of Ben Green problem by showing that<span><span><span><math><munder><mrow><mrow><mi>lim</mi></mrow><mspace></mspace><mrow><mi>sup</mi></mrow></mrow><mrow><mi>n</mi><mo>→</mo><mo>∞</mo></mrow></munder><mspace></mspace><mfrac><mrow><mfrac><mrow><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>16</mn></mrow></mfrac><msup><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>−</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub></mrow><mrow><mi>n</mi></mrow></mfrac><mo>≥</mo><mfrac><mrow><mi>π</mi></mrow><mrow><mn>4</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>0.193</mn><msup><mrow><mi>π</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>8</mn></mrow></mfrac><mo>.</mo></math></span></s","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114763"},"PeriodicalIF":0.7,"publicationDate":"2025-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144925287","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 connections between association schemes and analyses of polyhedral and positive semidefinite lift-and-project relaxations","authors":"Yu Hin Au Gary , Nathan Lindzey , Levent Tunçel","doi":"10.1016/j.disc.2025.114767","DOIUrl":"10.1016/j.disc.2025.114767","url":null,"abstract":"<div><div>We explore some connections between association schemes and the analyses of the semidefinite programming (SDP) based convex relaxations of combinatorial optimization problems in the Lovász–Schrijver lift-and-project hierarchy. Our analysis of the relaxations of the stable set polytope leads to bounds on the clique and stability numbers of some regular graphs reminiscent of classical bounds by Delsarte and Hoffman, as well as the notion of deeply vertex-transitive graphs — highly symmetric graphs that we show arise naturally from some association schemes. We also study relaxations of the hypergraph matching problem, and determine exactly or provide bounds on the lift-and-project ranks of these relaxations. Our proofs for these results also inspire the study of a homogeneous coherent configuration based on hypermatchings, which is an association scheme except it is generally non-commutative. We then illustrate the usefulness of obtaining commutative subschemes from non-commutative homogeneous coherent configurations via contraction in this context.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114767"},"PeriodicalIF":0.7,"publicationDate":"2025-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144925286","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 spectral radius of planar graphs with fixed size","authors":"Liangdong Fan , Liying Kang , Jiadong Wu","doi":"10.1016/j.disc.2025.114774","DOIUrl":"10.1016/j.disc.2025.114774","url":null,"abstract":"<div><div>Tait and Tobin (2017) <span><span>[18]</span></span> determined the unique spectral extremal graph over all outerplanar graphs and the unique spectral extremal graph over all planar graphs when the number of vertices is sufficiently large. In this paper we consider the spectral extremal problems of outerplanar graphs and planar graphs with a fixed number of edges. For planar graphs with <em>m</em> edges, our main result shows that the spectral extremal graph is the join of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>2</mn></mrow></msub></math></span> and <span><math><mfrac><mrow><mi>m</mi><mo>−</mo><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></math></span> isolated vertices when <em>m</em> is odd and sufficiently large, and the join of <span><math><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> and <span><math><msub><mrow><mi>S</mi></mrow><mrow><mfrac><mrow><mi>m</mi><mo>−</mo><mn>2</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msub><mo>∪</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>1</mn></mrow></msub></math></span> when <em>m</em> is even and sufficiently large.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"349 2","pages":"Article 114774"},"PeriodicalIF":0.7,"publicationDate":"2025-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144925196","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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